1) The document introduces trigonometric functions including sine, cosine, tangent, cotangent, secant and cosecant. It defines them based on angles and the coordinates of points on a unit circle.
2) Key properties of trigonometric functions are presented, including their periodicity and behavior under transformations like negation. Pythagorean identities relating sine, cosine and tangent are also proved.
3) Several examples are worked through to illustrate applying properties of trigonometric functions to solve equations and find all solutions within a given interval.
The document provides a math review covering topics in algebra, geometry, trigonometry, and statistics. It defines concepts like negative numbers, exponents, square roots, order of operations, lines, angles, trigonometric functions, and averages. Formulas are presented for topics like quadratic equations, the Pythagorean theorem, laws of sines and cosines, percentages, and standard deviation. Examples are included to illustrate key ideas.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
1. The document discusses techniques for finding extrema of functions, including absolute and local extrema. Critical points, endpoints, and the first and second derivative tests are covered.
2. The mean value theorem and Rolle's theorem are summarized. The mean value theorem relates the average and instantaneous rates of change over an interval.
3. Optimization problems can be solved by setting the derivative of the objective function equal to zero to find critical points corresponding to maxima or minima.
4. Newton's method is presented as an iterative process for approximating solutions to equations, using tangent lines to generate a sequence of improving approximations.
5. Anti-derivatives are defined as functions whose derivatives are a given
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
This document provides information about trigonometric ratios and their relationships to angles in the unit circle. It defines the trigonometric ratios of sine, cosine, and tangent. It discusses the four quadrants of the unit circle and identifies the signs of the trig ratios in each quadrant. Several examples are provided to illustrate calculating trig ratios based on the quadrant an angle falls in. The document also discusses graphs of the sine, cosine, and tangent functions over one period from 0 to 360 degrees. It provides exercises involving identifying quadrants, calculating trig ratios, relating angles in different quadrants, and solving trigonometric equations. Finally, it includes several past SPM exam questions involving applying trigonometric concepts in geometric contexts.
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
This document provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.
The document provides a math review covering topics in algebra, geometry, trigonometry, and statistics. It defines concepts like negative numbers, exponents, square roots, order of operations, lines, angles, trigonometric functions, and averages. Formulas are presented for topics like quadratic equations, the Pythagorean theorem, laws of sines and cosines, percentages, and standard deviation. Examples are included to illustrate key ideas.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
1. The document discusses techniques for finding extrema of functions, including absolute and local extrema. Critical points, endpoints, and the first and second derivative tests are covered.
2. The mean value theorem and Rolle's theorem are summarized. The mean value theorem relates the average and instantaneous rates of change over an interval.
3. Optimization problems can be solved by setting the derivative of the objective function equal to zero to find critical points corresponding to maxima or minima.
4. Newton's method is presented as an iterative process for approximating solutions to equations, using tangent lines to generate a sequence of improving approximations.
5. Anti-derivatives are defined as functions whose derivatives are a given
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
This document provides information about trigonometric ratios and their relationships to angles in the unit circle. It defines the trigonometric ratios of sine, cosine, and tangent. It discusses the four quadrants of the unit circle and identifies the signs of the trig ratios in each quadrant. Several examples are provided to illustrate calculating trig ratios based on the quadrant an angle falls in. The document also discusses graphs of the sine, cosine, and tangent functions over one period from 0 to 360 degrees. It provides exercises involving identifying quadrants, calculating trig ratios, relating angles in different quadrants, and solving trigonometric equations. Finally, it includes several past SPM exam questions involving applying trigonometric concepts in geometric contexts.
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
This document provides a summary of key concepts in algebra, geometry, trigonometry and their definitions. It includes formulas and properties for lines, polynomials, exponents, trig functions, triangles, circles, spheres, cones, cylinders, distance and the quadratic formula. Key topics covered are factoring, binomials, slope-intercept form, trig ratios, trig identities, trig reciprocals and the Pythagorean identities.
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
This document discusses general solutions for calculating stresses in beams with arbitrary cross-sections subjected to unsymmetrical bending. It presents equations to determine the neutral axis angle and stresses as a function of the bending moments and cross-sectional properties. As an example, it provides the solution for an L-shaped beam subjected to an inclined force, calculating the centroid and determining the maximum tensile and compressive stresses.
Rational functions are functions of the form f(x)=polynomial/polynomial. There are six key aspects to analyze in rational functions: y-intercept, x-intercepts, vertical asymptotes, horizontal/slant asymptotes, and the graph. Vertical asymptotes occur when the denominator is 0, x-intercepts when the numerator is 0, and horizontal/slant asymptotes depend on the relative degrees of the numerator and denominator polynomials.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
This document defines and provides examples of parameterized curves in R3. It introduces the concept of a smooth curve as a smooth map from an interval to R3. Regular curves are defined as smooth curves with everywhere nonzero velocity. Examples of regular curves include circles and helices parameterized by t. Reparameterizations are introduced as changing the parameterization of a curve while tracing the same path. The length of a curve is defined using an integral of speed over the parameter. Arc length parameterization reparameterizes a curve so that the parameter is equal to arc length measured from a fixed point.
Applied Calculus Chapter 1 polar coordinates and vectorJ C
The document discusses polar coordinates and vectors. It introduces parametric equations to describe the motion of a particle in the xy-plane over time. The variable t is called the parameter. Examples are provided to demonstrate forming Cartesian equations by eliminating t from parametric equations and graphing parametric equations by plugging in values of t. The document also discusses standard representation and finding direction numbers of vectors in R3.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
This document defines and provides examples of curves and parametrized curves. It discusses regular and unit-speed curves. The key points are:
i) A parametrized curve is a continuous function from an interval to Rn. Examples of parametrized curves include ellipses, parabolas, and helices.
ii) A regular curve is one where the derivative of the parametrization is never zero. A unit-speed curve has a derivative of constant length 1.
iii) The arc-length of a curve is defined as the integral of the derivative of the parametrization. Any reparametrization of a regular curve is also regular. A curve has a unit-
This document introduces Fourier series and their motivation. Joseph Fourier realized that many physical phenomena, like heat diffusion, could be modeled using partial differential equations. He developed a method of separation of variables to solve these equations, which led him to represent functions as infinite sums of sines and cosines, now known as Fourier series. The document outlines Fourier's approach, showing how assuming a solution of the form of separated variables leads to an eigenvalue problem. The eigenfunctions form a basis to represent more general functions as Fourier series. The Fourier coefficients that define a particular function can be determined by integrating the function against the eigenfunctions.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
The document provides information about graphing quadratic functions in standard form (y=ax^2 + bx + c). It discusses that the graph is a parabola that opens up or down depending on whether a is positive or negative. It also explains that the line of symmetry for the parabola is given by x=-b/2a and the vertex is found by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a quadratic function in standard form using three steps: finding the line of symmetry, finding the vertex, and finding/reflecting two other points to connect with a smooth curve.
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have a slope of 0 or undefined, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their characteristics like slope and intercept points.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
1) The document provides review material for Exam 3 of Math 285 including definitions of piecewise continuous and piecewise smooth functions and the convergence theorem for Fourier series.
2) It also defines Fourier series for periodic piecewise continuous functions and discusses Fourier sine and cosine series for odd and even functions.
3) Applications of Fourier series include using them to find formal solutions to boundary value problems involving differential equations.
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
The document discusses the properties of a rectangular hyperbola. A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular to each other. It has the equation x2/a2 - y2/a2 = 1. The eccentricity of a rectangular hyperbola is 2. Rotating the curve 45 degrees anticlockwise transforms it such that the asymptotes align with the x- and y-axes. The foci of the hyperbola are located at (a,a) and (-a,-a). The directrices are parallel to the y-axis and located at x = ±a.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
This document discusses Fourier series and their applications. It contains the following key points:
1. Fourier introduced Fourier series to solve heat equations through metal plates, expressing functions as infinite sums of sines and cosines.
2. Sine and cosine functions are orthogonal and periodic, allowing any piecewise continuous periodic function to be represented by a Fourier series.
3. The Euler-Fourier formulas relate the Fourier coefficients to the function, allowing the coefficients to be determined.
4. Even functions only have cosine terms, odd only sine, and the Fourier series converges to the average at discontinuities for piecewise continuous functions.
Este documento habla sobre los signos del Día del Juicio Final y cómo afectan la rectificación del corazón. El autor, Khaled Alazhari, discute varios signos como guerras, terremotos y desastres naturales. También explica cómo estos signos deberían motivar a las personas a mejorar sus acciones y acercarse más a Dios.
Este documento habla sobre los signos del Día del Juicio Final y cómo afectan la rectificación del corazón. El autor, Khaled Alazhari, discute varios signos como guerras, terremotos y desastres naturales. También explica cómo estos signos deberían motivar a las personas a mejorar sus acciones y acercarse más a Dios.
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
This document discusses general solutions for calculating stresses in beams with arbitrary cross-sections subjected to unsymmetrical bending. It presents equations to determine the neutral axis angle and stresses as a function of the bending moments and cross-sectional properties. As an example, it provides the solution for an L-shaped beam subjected to an inclined force, calculating the centroid and determining the maximum tensile and compressive stresses.
Rational functions are functions of the form f(x)=polynomial/polynomial. There are six key aspects to analyze in rational functions: y-intercept, x-intercepts, vertical asymptotes, horizontal/slant asymptotes, and the graph. Vertical asymptotes occur when the denominator is 0, x-intercepts when the numerator is 0, and horizontal/slant asymptotes depend on the relative degrees of the numerator and denominator polynomials.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
This document defines and provides examples of parameterized curves in R3. It introduces the concept of a smooth curve as a smooth map from an interval to R3. Regular curves are defined as smooth curves with everywhere nonzero velocity. Examples of regular curves include circles and helices parameterized by t. Reparameterizations are introduced as changing the parameterization of a curve while tracing the same path. The length of a curve is defined using an integral of speed over the parameter. Arc length parameterization reparameterizes a curve so that the parameter is equal to arc length measured from a fixed point.
Applied Calculus Chapter 1 polar coordinates and vectorJ C
The document discusses polar coordinates and vectors. It introduces parametric equations to describe the motion of a particle in the xy-plane over time. The variable t is called the parameter. Examples are provided to demonstrate forming Cartesian equations by eliminating t from parametric equations and graphing parametric equations by plugging in values of t. The document also discusses standard representation and finding direction numbers of vectors in R3.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
This document defines and provides examples of curves and parametrized curves. It discusses regular and unit-speed curves. The key points are:
i) A parametrized curve is a continuous function from an interval to Rn. Examples of parametrized curves include ellipses, parabolas, and helices.
ii) A regular curve is one where the derivative of the parametrization is never zero. A unit-speed curve has a derivative of constant length 1.
iii) The arc-length of a curve is defined as the integral of the derivative of the parametrization. Any reparametrization of a regular curve is also regular. A curve has a unit-
This document introduces Fourier series and their motivation. Joseph Fourier realized that many physical phenomena, like heat diffusion, could be modeled using partial differential equations. He developed a method of separation of variables to solve these equations, which led him to represent functions as infinite sums of sines and cosines, now known as Fourier series. The document outlines Fourier's approach, showing how assuming a solution of the form of separated variables leads to an eigenvalue problem. The eigenfunctions form a basis to represent more general functions as Fourier series. The Fourier coefficients that define a particular function can be determined by integrating the function against the eigenfunctions.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
The document provides information about graphing quadratic functions in standard form (y=ax^2 + bx + c). It discusses that the graph is a parabola that opens up or down depending on whether a is positive or negative. It also explains that the line of symmetry for the parabola is given by x=-b/2a and the vertex is found by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a quadratic function in standard form using three steps: finding the line of symmetry, finding the vertex, and finding/reflecting two other points to connect with a smooth curve.
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have a slope of 0 or undefined, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their characteristics like slope and intercept points.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
1) The document provides review material for Exam 3 of Math 285 including definitions of piecewise continuous and piecewise smooth functions and the convergence theorem for Fourier series.
2) It also defines Fourier series for periodic piecewise continuous functions and discusses Fourier sine and cosine series for odd and even functions.
3) Applications of Fourier series include using them to find formal solutions to boundary value problems involving differential equations.
1. The document discusses the concept of derivatives and how to calculate them. It defines key terms like increment, average rate of change, and instantaneous rate of change.
2. Methods are provided for calculating the derivative of various types of functions, including polynomials, rational functions, and functions with roots. Examples are worked through step-by-step.
3. The relationship between derivatives and tangent lines is explained, with derivatives defined as the instantaneous rate of change or slope of the curve at a point.
The document discusses the properties of a rectangular hyperbola. A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular to each other. It has the equation x2/a2 - y2/a2 = 1. The eccentricity of a rectangular hyperbola is 2. Rotating the curve 45 degrees anticlockwise transforms it such that the asymptotes align with the x- and y-axes. The foci of the hyperbola are located at (a,a) and (-a,-a). The directrices are parallel to the y-axis and located at x = ±a.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
This document discusses Fourier series and their applications. It contains the following key points:
1. Fourier introduced Fourier series to solve heat equations through metal plates, expressing functions as infinite sums of sines and cosines.
2. Sine and cosine functions are orthogonal and periodic, allowing any piecewise continuous periodic function to be represented by a Fourier series.
3. The Euler-Fourier formulas relate the Fourier coefficients to the function, allowing the coefficients to be determined.
4. Even functions only have cosine terms, odd only sine, and the Fourier series converges to the average at discontinuities for piecewise continuous functions.
Este documento habla sobre los signos del Día del Juicio Final y cómo afectan la rectificación del corazón. El autor, Khaled Alazhari, discute varios signos como guerras, terremotos y desastres naturales. También explica cómo estos signos deberían motivar a las personas a mejorar sus acciones y acercarse más a Dios.
Este documento habla sobre los signos del Día del Juicio Final y cómo afectan la rectificación del corazón. El autor, Khaled Alazhari, discute varios signos como guerras, terremotos y desastres naturales. También explica cómo estos signos deberían motivar a las personas a mejorar sus acciones y acercarse más a Dios.
The document summarizes key concepts about functions and lines. It defines a function as a rule that assigns unique output values to inputs. Functions can be represented graphically by plotting the points (x, f(x)). Lines on a plane can be defined by two points and have a slope that represents the rise over run. The slope formula is used to find the equation of a line in y=mx+b form, where m is the slope and b is the y-intercept. Perpendicular and parallel lines have specific relationships between their slopes.
What is the subject ?: To solve one of the youth’ problems in the west
……………………………….
Who are we ? Parents and imams
why ? To learn how to deal with youth
Where ? Islamic center in Sydney
When ? Today
10 factors for uniting muslims in australiaImam Al Azhari
This document outlines 10 factors for uniting Muslims in the West:
1. Ikhlas or sincerity in worshipping Allah alone.
2. Holding fast to the Quran and Sunnah (teachings of Prophet Muhammad).
3. Understanding that Muslim unity is obligatory, not optional.
4. Supporting qualified imams and wise leaders.
5. Tolerating and accepting one another.
6. Spreading brotherhood and love among Muslims.
7. Reflecting on Islamic rituals like prayer, zakat, and Hajj.
8. Making dua (supplication) for unity.
The document emphasizes that
How to follow the Prophet Mohammed (PBUH) and glorify his Sunnah
Alatba’a is a must
Six conditions for following the prophet’s acts of worship
The benefits of following Sunna
Symptoms of Following our prophet
1) The document provides a lesson on cellular respiration that uses simulations to explore the pathways of glycolysis, the Krebs cycle, and the electron transport chain.
2) Students will learn about the key molecules and enzymes involved in each pathway, how they extract energy from glucose, and the relative energetic content of molecules like ATP.
3) The lesson assesses student understanding with embedded questions and is designed to be used with TI-Nspire technology for interactive simulations and monitoring of student progress.
This course syllabus outlines the topics, schedule, and requirements for a graduate level Molecular and Cellular Bioengineering course. Over 50 contact hours, the course will cover topics such as DNA engineering, gene regulation, molecular and cellular techniques, cellular functions, cell-environment interactions, tissue engineering, and advanced topics including stem cell biology and fluorescence proteins/biosensors. Students will be evaluated based on five quizzes, a class presentation, and a final exam. Recommended textbooks and research papers are provided as additional references for each major topic.
1) The document discusses inequalities involving real numbers. It introduces basic rules for how inequalities are affected by addition, multiplication, taking squares, and taking reciprocals.
2) Several examples are provided to demonstrate how to solve various inequalities by applying these rules. For instance, it is shown that the inequality 4x + 7 < 3 is equivalent to x < -1.
3) Graphs are used to confirm the solutions obtained from manipulating the inequalities. The key results are that addition preserves inequalities while multiplication preserves or reverses them depending on whether the number being multiplied is positive or negative.
1. The document proposes a new approach called Network Inference with Pooling Data (NIPD) to learn condition-specific networks that identify both shared and unique patterns across conditions.
2. NIPD learns networks by pooling data from multiple conditions simultaneously, unlike previous approaches that learn networks independently per condition.
3. The authors apply NIPD to microarray data from yeast under different starvation conditions, finding both shared responses like respiration as well as condition-specific interactions, validating the approach.
This document discusses challenges faced by Muslim youth in the West and proposes solutions. It describes an example of a father whose 23-year old daughter wanted to stay out late, against his wishes. It then discusses that the main reasons for this issue are poor communication, late curfews without discussion, and lack of religious knowledge. The document proposes that the best time for teaching is from a young age through good parenting and discussions. It provides tips for effective communication skills and discusses using prophetic teaching methods of respect, advice and setting a good example. Finally, it outlines the key beliefs and pillars of Islam that should be taught to youth.
The document describes an individual who has 5 brothers and 1 sister. It focuses on the third oldest brother, Andrew, who is 7 years old and enjoys the TV show Super Hero Squad and the character The Hulk. The individual bought Andrew Hulk hands for Christmas that he loved. Andrew helps with younger brothers and agrees with the individual.
The document then switches topics to discuss different friends, family members, hobbies, TV shows and movies that the individual enjoys discussing. It provides snippets of information about each topic in short paragraphs without much connection between the topics.
Este documento proporciona instrucciones sobre cómo crear informes en una base de datos, incluyendo seleccionar los campos de datos, establecer un orden y distribución de la información, agregar un título, y generar una vista previa del informe.
El documento describe los pasos para crear un juego de triqui en Excel, incluyendo iniciar un nuevo documento de Excel, formato la cuadrícula a 48 celdas, centrar el contenido horizontal y verticalmente, aplicar validación de datos para celdas X y O, copiar fórmulas para completar la cuadrícula, y contar las posibilidades de victoria para cada jugador.
Este documento proporciona instrucciones para crear formularios en Access que permitan modificar tablas de datos. Explica que primero se debe abrir Access y seleccionar "Asistente para formularios", luego elegir la tabla a la que se le hará el formulario y agregar todos sus campos. Además, recomienda dejar la opción "En columnas" y configurar el formulario con un encabezado, detalles y pie de página. Finalmente, indica repetir este proceso para las otras cuatro tablas.
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1. ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c W W L Chen, X T Duong and Macquarie University, 1999.
This work is available free, in the hope that it will be useful.
Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or any information storage and retrieval system, with or without permission from the authors.
Chapter 3
TRIGONOMETRY
3.1. Radian and Arc Length
The number π plays a central role in the study of trigonometry. We all know that a circle of radius 1
has area π and circumference 2π. It is also very useful in describing angles, as we shall now show.
Let us split a circle of radius 1 along a diameter into two semicircles as shown in the picture below.
The circumference of the circle is now split into two equal parts, each of length π and each suntending
an angle 180◦ . If we use the convention that π = 180◦ , then the arc of the semicircle of radius 1 will be
the same as the angle it subtends. If we further split the arc of the semicircle of radius 1 into two equal
parts, then each of the two parts forms an arc of length π/2 and subtends an angle 90◦ = π/2. In fact,
any arc of a circle of radius 1 which subtends an angle θ must have length θ under our convention.
We now formalize our discussion so far.
Definition. An angle of 1 radian is defined to be the angle subtended by an arc of length 1 on a circle
of radius 1.
† This chapter was written at Macquarie University in 1999.
2. 3–2 W W L Chen and X T Duong : Elementary Mathematics
Remarks. (1) Very often, the term radian is omitted when we discuss angles. We simply refer to an
angle 1 or an angle π, rather than an angle of 1 radian or an angle of π radian.
(2) Simple calculation shows that 1 radian is equal to (180/π)◦ = 57.2957795 . . .◦ . Similarly, we
can show that 1◦ is equal to (π/180) radian = 0.01745329 . . . radian. In fact, since π is irrational, the
digits do not terminate or repeat.
(3) We observe the following special values:
π π π π
= 30◦ , = 45◦ , = 60◦ , = 90◦ , π = 180◦ , 2π = 360◦ .
6 4 3 2
Consider now a circle of radius r and an angle θ given in radian, as shown in the picture below.
rθ
θ
r
Clearly the length s of the arc which subtends the angle θ satisfies s = rθ, while the area A of the sector
satisfies
θ 1
A = πr2 × = r2 θ.
2π 2
Note that πr2 is equal to the area inside the circle, while θ/2π is the proportion of the area in question.
3.2. The Trigonometric Functions
Consider the xy-plane, together with a circle of radius 1 and centred at the origin (0, 0). Suppose that
θ is an angle measured anticlockwise from the positive x-axis, and the point (x, y) on the circle is as
shown in the picture below.
y
1
θ
x
We define
cos θ = x and sin θ = y.
Furthermore, we define
sin θ y cos θ x 1 1 1 1
tan θ = = , cot θ = = , sec θ = = and csc θ = = .
cos θ x sin θ y cos θ x sin θ y
3. Chapter 3 : Trigonometry 3–3
Remarks. (1) Note that tan θ and sec θ are defined only when cos θ = 0, and that cot θ and csc θ are
defined only when sin θ = 0.
(2) It is a good habit to always measure an angle from the positive x-axis, using the convention that
positive angles are measured anticlockwise and negative angles are measured clockwise, as illustrated
below.
1
3π
4
-1
π
2
(3) We then observe that
horizontal side vertical side
cos θ = and sin θ = ,
hypothenuse hypothenuse
as well as
vertical side horizontal side
tan θ = and cot θ = ,
horizontal side vertical side
with the convention that
> 0 (to the right of the (vertical) y-axis),
horizontal side
< 0 (to the left of the (vertical) y-axis),
and
> 0 (above the (horizontal) x-axis),
vertical side
< 0 (below the (horizontal) x-axis),
while
hypothenuse > 0 (always).
(4) It is also useful to remember the CAST rule concerning sine, cosine and tangent.
S (sin > 0) A (all > 0)
T (tan > 0) C (cos > 0)
PYTHAGOREAN IDENTITIES. For every value θ ∈ R for which the trigonometric functions in
question are defined, we have
(a) cos2 θ + sin2 θ = 1;
(b) 1 + tan2 θ = sec2 θ; and
(c) 1 + cot2 θ = csc2 θ.
4. 3–4 W W L Chen and X T Duong : Elementary Mathematics
Proof. Note that (a) follows from the classical Pythagoras’s theorem. If cos θ = 0, then dividing both
sides of (a) by cos2 θ gives (b). If sin θ = 0, then dividing both sides of (a) by sin2 θ gives (c). ♣
We sketch the graphs of the functions y = sin x, y = cos x and y = tan x for −2π ≤ x ≤ 2π below:
y
1
y = sin x
x
-2π -π π 2π
-1
y
1
y = cos x
x
-2π -π π 2π
-1
y
y = tan x
x
-2π -π π 2π
We shall use these graphs to make some observations about trigonometric functions.
5. Chapter 3 : Trigonometry 3–5
PROPERTIES OF TRIGONOMETRIC FUNCTIONS.
(a) The functions sin x and cos x are periodic with period 2π. More precisely, for every x ∈ R, we have
sin(x + 2π) = sin x and cos(x + 2π) = cos x.
(b) The functions tan x and cot x are periodic with period π. More precisely, for every x ∈ R for which
the trigonometric function in question is defined, we have
tan(x + π) = tan x and cot(x + π) = cot x.
(c) The function sin x is an odd function, and the function cos x is an even function. More precisely,
for every x ∈ R, we have
sin(−x) = − sin x and cos(−x) = cos x.
(d) The functions tan x and cot x are odd functions. More precisely, for every x ∈ R for which the
trigonometric function in question is defined, we have
tan(−x) = − tan x and cot(−x) = − cot x.
(e) For every x ∈ R, we have
sin(x + π) = − sin x and cos(x + π) = − cos x.
(f) For every x ∈ R, we have
sin(π − x) = sin x and cos(π − x) = − cos x.
(g) For every x ∈ R for which the trigonometric function in question is defined, we have
tan(π − x) = − tan x and cot(π − x) = − cot x.
(h) For every x ∈ R, we have
π π
sin x + = cos x and cos x + = − sin x.
2 2
(i) For every x ∈ R, we have
π π
sin − x = cos x and cos − x = sin x.
2 2
(j) For every x ∈ R for which the trigonometric functions in question are defined, we have
π π
tan x + = − cot x and cot x + = − tan x.
2 2
(k) For every x ∈ R for which the trigonometric function in question is defined, we have
π π
tan − x = cot x and cot − x = tan x.
2 2
Remark. There is absolutely no need to remember any of these properties! We shall discuss later
some trigonometric identities which will give all the above as special cases.
6. 3–6 W W L Chen and X T Duong : Elementary Mathematics
Example 3.2.1. Consider the following picture.
1
π/6
Clearly the triangle shown is an equilateral triangle, with all three sides of equal length. It is also clear
that sin(π/6) is half the length of the vertical side. It follows that we must have sin(π/6) = 1/2. To find
the precise value for cos(π/6), we first observe that cos(π/6) > 0. On the other hand, it follows from the
√
first of the Pythagorean identities that cos2 (π/6) = 3/4. Hence cos(π/6) = 3/2. We can then deduce
√
that tan(π/6) = 1/ 3.
√
Example 3.2.2. We have tan(13π/6) = tan(7π/6) = tan(π/6) = 1/ 3. Note that we have used part
(b) of the Properties of trigonometric functions, as well as the result from Example 3.2.1.
Example 3.2.3. Consider the following picture.
-π/3
1
Clearly the triangle shown is an equilateral triangle, with all three sides of equal length. It is also clear
that cos(−π/3) is half the length the horizontal side. It follows that we must have cos(−π/3) = 1/2.
To find the precise value for sin(−π/3), we first observe that sin(−π/3) < 0. On the other hand,
follows from the first of the Pythagorean identities that sin2 (−π/3) = 3/4. Hence sin(−π/3) =
it √
− 3/2. Alternatively, we can deduce from Example 3.2.1 by using parts (c) and (i) of the Properties of
7. Chapter 3 : Trigonometry 3–7
trigonometric functions that
π π π π π 1
cos − = cos = sin − = sin =
3 3 2 3 6 2
and √
π π π π π 3
sin − = − sin = − cos − = − cos = − .
3 3 2 3 6 2
√
We can then deduce that tan(−π/3) = − 3.
Example 3.2.4. Consider the following picture.
5 π/4
1
Clearly the triangle shown is a right-angled triangle with the two shorter sides of equal length and
hypothenuse of length 1. It is also clear that sin(5π/4) is the length of the vertical side, with a − sign
attached as it is below the horizontal axis. On the other hand, it is also clear that cos(5π/4) is the
length of the horizontal side, again with a − sign attached as it is left of the vertical axis. It follows
from Pythagoras’s theorem that sin(5π/4) = cos(5π/4) = y, where y < 0 and y 2 + y 2 = 1. Clearly
√
y = −1/ 2. We can then deduce that tan(5π/4) = 1.
√
Example 3.2.5. To find all the solutions of the equation sin x = −1/ 2 in the interval 0 ≤ x < 2π,
we consider the following picture.
7 π/4
1 1
8. 3–8 W W L Chen and X T Duong : Elementary Mathematics
Using Pythagoras’s theorem, it is easy to see that the two triangles shown both have horizontal side of
√
length 1/ 2, the same as the length of their vertical sides. Clearly x = 5π/4 or x = 7π/4.
Example 3.2.6. Suppose that we wish to find all the solutions of the equation cos2 x = 1/4 in the
interval −π/2 < x ≤ π. Observe first of all that either cos x = 1/2 or cos x = −1/2. We consider the
following picture.
1 1
2 π/3
- π/3
1
Using Pythagoras’s theorem, it is easy to see that the three triangles shown both have vertical side of
√
length 3/2. Clearly x = −π/3, x = π/3 or x = 2π/3.
Example 3.2.7. Convince yourself that the only two solutions of the equation cos2 x = 1 in the interval
0 ≤ x < 2π are x = 0 and x = π.
√
Example 3.2.8. To find all the solutions of the equation tan x = 3 in the interval 0 ≤ x < 2π, we
consider the following picture.
1
4π/3
1
Since tan x > 0, it follows from the CAST rule that we can restrict our attention to the first and third
quadrants. It is easy to check that the two triangles shown have horizontal sides of length 1/2 and
√
vertical sides of length 3/2. Clearly x = π/3 or x = 4π/3.
9. Chapter 3 : Trigonometry 3–9
Example 3.2.9. Suppose that we wish to find all the solutions of the equation √ 2 x = 1/3 in the
√ tan
interval 0 ≤ x ≤ 3π/2. Observe first of all that either tan x = 1/ 3 or tan x = −1/ 3. We consider the
following picture.
1 7 π/6 1
1
√
It is easy to check that the three triangles shown have horizontal sides of length 3/2 and vertical sides
of length 1/2. Clearly x = π/6, x = 5π/6 or x = 7π/6.
√
Example 3.2.10. Convince yourself that the only two solutions of the equation sec x = 2 in the
interval 0 ≤ x < 2π are x = π/4 and x = 7π/4.
Example 3.2.11. For every x ∈ R, we have
sin3 x + sin x cos2 x = sin x sin2 x + sin x cos2 x = (sin x)(sin2 x + cos2 x) = sin x,
in view of the first of the Pythagorean identities.
Example 3.2.12. For every x ∈ R such that cos x = 0, we have
(sec x − tan x)(sec x + tan x) = sec2 x − tan2 x = 1,
in view of the second of the Pythagorean identities.
Example 3.2.13. For every x ∈ R such that cos x = ±1, we have
1 1 (1 + cos x) + (1 − cos x) 2 2
+ = = = = 2 csc2 x.
1 − cos x 1 + cos x (1 − cos x)(1 + cos x) 1 − cos2 x sin2 x
Example 3.2.14. For every x ∈ R, we have
(cos x + sin x)2 + (cos x − sin x)2 = (cos2 x + 2 cos x sin x + sin2 x) + (cos2 x − 2 cos x sin x + sin2 x)
= (1 + 2 cos x sin x) + (1 − 2 cos x sin x) = 2.
Example 3.2.15. For every x ∈ R such that the expression on the left hand side makes sense, we have
1 + cot x 1 + tan x cos x sin x
− = (1 + cot x) sin x − (1 + tan x) cos x = 1+ sin x − 1 + cos x
csc x sec x sin x cos x
= (sin x + cos x) − (cos x + sin x) = 0.
10. 3–10 W W L Chen and X T Duong : Elementary Mathematics
Example 3.2.16. Let us return to Example 3.2.5 where we showed that the solutions of the equation
√
sin x = −1/ 2 in the interval 0 ≤ x < 2π are given by x = 5π/4 and x = 7π/4. Suppose now that we
wish to find all the values x ∈ R that satisfy the same equation. To do this, we can use part (a) of the
Properties of trigonometric functions, and conclude that the solutions are given by
5π 7π
x= + 2kπ or x= + 2kπ,
4 4
where k ∈ Z.
Example 3.2.17. Consider the equation sec(x/2) = 2. Then cos(x/2) = 1/2. If we first restrict our
attention to 0 ≤ x/2 < 2π, then it is not difficult to see that the solutions are given by x/2 = π/3 and
x/2 = 5π/3. Using part (a) of the Properties of trigonometric functions, we conclude that without the
restriction 0 ≤ x/2 < 2π, the solutions are given by
x π x 5π
= + 2kπ or = + 2kπ,
2 3 2 3
where k ∈ Z. It follows that
2π 10π
x= + 4kπ or x= + 4kπ,
3 3
where k ∈ Z.
3.3. Some Trigonometric Identities
Consider a triangle with side lengths and angles as shown in the picture below:
J
JJJ
A J
JJJJ
JJJJ
JJJb
c JJJ
JJJJ
JJJJ
JJJJ
JJJ
B C
a
SINE RULE. We have
a b c
= = .
sin A sin B sin C
COSINE RULE. We have
a2 = b2 + c2 − 2bc cos A,
b2 = a2 + c2 − 2ac cos B,
c2 = a2 + b2 − 2ab cos C.
Sketch of Proof. Consider the picture below.
J
JJJJ
JJJJ
JJJJ
JJJb
c JJJ
JJJJ
JJJJ
JJJJ
JJJ
B C
o /
a
11. Chapter 3 : Trigonometry 3–11
Clearly the length of the vertical line segment is given by c sin B = b sin C, so that
b c
= .
sin B sin C
This gives the sine rule. Next, note that the horizontal side of the the right-angled triangle on the left
has length c cos B. It follows that the horizontal side of the right-angled triangle on the right has length
a − c cos B. If we now apply Pythagoras’s theorem to this latter triangle, then we have
(a − c cos B)2 + (c sin B)2 = b2 ,
so that
a2 − 2ac cos B + c2 cos2 B + c2 sin2 B = b2 ,
whence
b2 = a2 + c2 − 2ac cos B.
This gives the cosine rule. ♣
We mentioned earlier that there is no need to remember any of the Properties of trigonometric
functions discussed in the last section. The reason is that they can all be deduced easily from the
identities below.
SUM AND DIFFERENCE IDENTITIES. For every A, B ∈ R, we have
sin(A + B) = sin A cos B + cos A sin B and sin(A − B) = sin A cos B − cos A sin B,
as well as
cos(A + B) = cos A cos B − sin A sin B and cos(A − B) = cos A cos B + sin A sin B.
Remarks. (1) Proofs can be sketched for these identities by drawing suitable pictures, although such
pictures are fairly complicated. We omit the proofs here.
(2) It is not difficult to remember these identities. Observe the pattern
sin ± = sin cos ± cos sin and cos ± = cos cos ∓ sin sin .
For small positive angles, increasing the angle increases the sine (thus keeping signs) and decreases the
cosine (thus reversing signs).
(3) One can also deduce analogous identities for tangent and cotangent. We have
sin(A + B) sin A cos B + cos A sin B
tan(A + B) = = .
cos(A + B) cos A cos B − sin A sin B
Dividing both the numerator and denominator by cos A cos B, we obtain
sin A sin B
+
cos A cos B tan A + tan B
tan(A + B) = = .
sin A sin B 1 − tan A tan B
1−
cos A cos B
Similarly, one can deduce that
tan A − tan B
tan(A − B) = .
1 + tan A tan B
12. 3–12 W W L Chen and X T Duong : Elementary Mathematics
Example 3.3.1. For every x ∈ R, we have
π π π
sin − x = sin cos x − cos sin x = cos x,
2 2 2
and
π π π
cos − x = cos cos x + sin sin x = sin x.
2 2 2
These form part (i) of the Properties of trigonometric functions.
Of particular interest is the special case when A = B.
DOUBLE ANGLE IDENTITIES. For every x ∈ R, we have
(a) sin 2x = 2 sin x cos x; and
(b) cos 2x = cos2 x − sin2 x = 1 − 2 sin2 x = 2 cos2 x − 1.
HALF ANGLE IDENTITIES. For every y ∈ R, we have
y 1 − cos y y 1 + cos y
sin2 = and cos2 = .
2 2 2 2
Proof. Let x = y/2. Then part (b) of the Double angle identities give
y y
cos y = 1 − 2 sin2 = 2 cos2 − 1.
2 2
The results follow easily. ♣
Example 3.3.2. Suppose that we wish to find the precise values of cos(−3π/8) and sin(−3π/8). We
have
3π 1 + cos(−3π/4)
cos2 − = .
8 2
√
It is not difficult to show that cos(−3π/4) = −1/ 2, so that
√
3π 1 1 2−1
cos 2
− = 1− √ = √ .
8 2 2 2 2
It is easy to see that cos(−3π/8) > 0, and so
√
3π 2−1
cos − = √ .
8 2 2
Similarly, we have
√
3π 1 − cos(−3π/4) 1 1 2+1
sin2
− = = 1+ √ = √ .
8 2 2 2 2 2
It is easy to see that sin(−3π/8) < 0, and so
√
3π 2+1
sin − =− √ .
8 2 2
13. Chapter 3 : Trigonometry 3–13
Example 3.3.3. Suppose that α is an angle in the first quadrant and β is an angle in the third
quadrant. Suppose further that sin α = 3/5 and cos β = −5/13. Using the first of the Pythagorean
identities, we have
16 144
cos2 α = 1 − sin2 α = and sin2 β = 1 − cos2 β = .
25 169
On the other hand, using the CAST rule, we have cos α > 0 and sin β < 0. It follows that cos α = 4/5
and sin β = −12/13. Then
3 5 4 12 33
sin(α − β) = sin α cos β − cos α sin β = × − − × − =
5 13 5 13 65
and
4 5 3 12 56
cos(α − β) = cos α cos β + sin α sin β = × − + × − =− ,
5 13 5 13 65
so that
sin(α − β) 33
tan(α − β) = =− .
cos(α − β) 56
Example 3.3.4. For every x ∈ R, we have
sin 6x cos 2x − cos 6x sin 2x = sin(6x − 2x) = sin 4x = 2 sin 2x cos 2x.
Note that the first step uses a difference identity, while the last step uses a double angle identity.
Example 3.3.5. For appropriate values of α, β ∈ R, we have
sin(α + β) sin α cos β + cos α sin β 1 + cot α tan β
= = .
sin(α − β) sin α cos β − cos α sin β 1 − cot α tan β
Note that the first step uses sum and difference identities, while the second step involves dividing both
the numerator and the denominator by sin α cos β.
Example 3.3.6. For every x ∈ R for which sin 4x = 0, we have
cos 8x cos2 4x − sin2 4x
2 = = cot2 4x − 1.
sin 4x sin2 4x
Note that the first step involves a double angle identity.
Example 3.3.7. For every x ∈ R for which cos x = 0, we have
sin 2x 2 sin x cos x
= = 2 tan x.
1 − sin2 x cos2 x
Note that the first step involves a double angle identity as well as a Pythagorean identity.
Example 3.3.8. For every x ∈ R for which sin 3x = 0 and cos 3x = 0, we have
cos2 3x − sin2 3x cos 6x
= = cot 6x.
2 sin 3x cos 3x sin 6x
Note that the first step involves double angle identities.
14. 3–14 W W L Chen and X T Duong : Elementary Mathematics
Example 3.3.9. This example is useful in calculus for finding the derivatives of the sine and cosine
functions. For every x, h ∈ R with h = 0, we have
sin(x + h) − sin x sin x cos h + cos x sin h − sin x sin h cos h − 1
= = (cos x) × + (sin x) ×
h h h h
and
cos(x + h) − cos x cos x cos h − sin x sin h − cos x sin h cos h − 1
= = −(sin x) × + (cos x) × .
h h h h
When h is very close to 0, then
sin h cos h − 1
≈1 and ≈ 0,
h h
so that
sin(x + h) − sin x cos(x + h) − cos x
≈ cos x and ≈ − sin x.
h h
This is how we show that the derivatives of sin x and cos x are respectively cos x and − sin x.
Problems for Chapter 3
1. Find the precise value of each of the following quantities, showing every step of your argument:
4π 4π 3π 3π
a) sin b) tan c) cos d) tan
3 3 4 4
65π 47π 25π 37π
e) tan f) sin − g) cos h) cot −
4 6 3 2
5π 5π 53π 53π
i) sin j) tan k) cos − l) cot −
3 3 6 6
2. Find all solutions for each of the following equations in the intervals given, showing every step of
your argument:
1 1
a) sin x = − , 0 ≤ x < 2π b) sin x = − , 0 ≤ x < 4π
2 2
1 1
c) sin x = − , −π ≤ x < π d) sin x = − , 0 ≤ x < 3π
2 2
3 3
e) cos2 x = , 0 ≤ x < 2π f) cos2 x = , −π ≤ x < 3π
4 4
g) tan2 x = 3, 0 ≤ x < 2π h) tan2 x = 3, −2π ≤ x < π
5π
i) cot2 x = 3, 0 ≤ x < 2π j) sec2 x = 2, 0≤x<
2
1 1
k) cos x = − , 0 ≤ x < 2π l) cos x = − , 0 ≤ x < 4π
2 2
1 π
m) tan2 x = 1, 2π ≤ x < 4π n) sin2 x = , ≤ x < 2π
4 2
3. Simplify each of the following expressions, showing every step of yor argument:
sin x sec x − sin2 x tan x
a) b) sin 3x cos x + sin x cos 3x
sin 2x
c) cos 5x cos x + sin x sin 5x d) sin 3x − cos 2x sin x − 2 sin x cos2 x
sin 4x
e) 2 x − sin2 x) sin x cos x
f) cos 4x cos 3x − 4 sin x sin 3x cos x cos 2x
(cos
g) 2 sin 3x cos 3x cos 5x − (cos2 3x − sin2 3x) sin 5x
15. Chapter 3 : Trigonometry 3–15
4. We know that
√
π 1 π 3 π π 1 5π 1 π π
sin = , cos = , sin = cos = √ and = + .
6 2 6 2 4 4 2 24 2 4 6
We also know that
cos(α + β) = cos α cos β − sin α sin β and cos 2θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ.
Use these to determine the exact values of
5π 5π
cos and sin .
24 24
[Hint: Put your √
√ calculators away. Your answers will be square roots of expressions involving the
numbers 2 and 3.]
5. Find the precise value of each of the following quantities, showing every step of your argument:
π π 7π 7π
a) cos − b) sin − c) cos d) tan
8 8 12 12
6. Use the sum and difference identities for the sine and cosine functions to deduce each of the following
identities:
a) sin(x + π) = − sin x b) cos(−x) = cos x c) tan(π − x) = − tan x
π
d) cot + x = − tan x
2
− ∗ − ∗ − ∗ − ∗ − ∗ −