The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths in different triangles.
The document discusses the sine rule and formulas for solving triangles. It introduces the sine rule, which states that in any triangle ABC, a/sinA = b/sinB = c/sinC. It then shows examples of using the sine rule to calculate unknown side lengths and heights of triangles. The document also covers the formula for calculating the area of a triangle as 1/2 * base * height or 1/2 * ab * sinC.
The document discusses trigonometric functions and radian measure. It states that 360 degrees equals 2 pi radians. It then provides a table with common conversions between degrees and radians for angles from 30 to 360 degrees. Examples are also given of converting degrees to radians and radians to degrees.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document defines terminology and notation used in Euclidean geometry. It provides definitions for terms like parallel, perpendicular, congruent, and similar. It also outlines conventions for naming angles, polygons, and parallel lines. The document concludes by describing the key rules and theorems involved in constructing geometric proofs, such as the fact that angles in a straight line sum to 180 degrees and that angles around a point sum to 360 degrees.
Here are the key points about true bearings:
- True bearings are always measured clockwise from North
- To find the bearing of point B from point A, start at North and measure the angle clockwise to the line from A to B
- The bearing of Y from X is 120° or S60°E
- The bearing of X from Y would be N60°W
So in summary, true bearings provide a unambiguous way to specify the direction from one point to another by always measuring clockwise from North.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths in different triangles.
The document discusses the sine rule and formulas for solving triangles. It introduces the sine rule, which states that in any triangle ABC, a/sinA = b/sinB = c/sinC. It then shows examples of using the sine rule to calculate unknown side lengths and heights of triangles. The document also covers the formula for calculating the area of a triangle as 1/2 * base * height or 1/2 * ab * sinC.
The document discusses trigonometric functions and radian measure. It states that 360 degrees equals 2 pi radians. It then provides a table with common conversions between degrees and radians for angles from 30 to 360 degrees. Examples are also given of converting degrees to radians and radians to degrees.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document defines terminology and notation used in Euclidean geometry. It provides definitions for terms like parallel, perpendicular, congruent, and similar. It also outlines conventions for naming angles, polygons, and parallel lines. The document concludes by describing the key rules and theorems involved in constructing geometric proofs, such as the fact that angles in a straight line sum to 180 degrees and that angles around a point sum to 360 degrees.
Here are the key points about true bearings:
- True bearings are always measured clockwise from North
- To find the bearing of point B from point A, start at North and measure the angle clockwise to the line from A to B
- The bearing of Y from X is 120° or S60°E
- The bearing of X from Y would be N60°W
So in summary, true bearings provide a unambiguous way to specify the direction from one point to another by always measuring clockwise from North.
The document discusses trigonometry concepts related to 3D shapes and solving problems involving angles of elevation. Specifically:
- When doing 3D trigonometry, it is often useful to redraw shapes in 2D to analyze them.
- A worked example problem is shown to find the distance and bearing between a life raft (David's position) and a search vessel (Anna's position) based on angles of elevation they each observe of a mountain peak.
- Applying trigonometric relationships involving angles and the mountain's known height, the distance between David and Anna is calculated to be 2799 meters, and the bearing of David from Anna is calculated to be 249°51'.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, eliminating variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of variables and equations can always be solved using this method.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its direction. It is solved as the number if it is positive, and the opposite of the number if it is negative. Examples of evaluating absolute value expressions are provided. The document also discusses solving absolute value equations and inequalities, providing examples of setting up and solving different absolute value equations for the variable.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses coordinate geometry concepts including the distance formula and midpoint formula. It explains that the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint. It also discusses dividing intervals, noting that for a level 2 math exam it is restricted to midpoint divisions in a 1:1 ratio, while an extension 1 exam can involve any ratio for internal or external divisions. Examples are provided to illustrate the concepts.
The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope:
(1) The rise over the run between two points (vertical change over horizontal change)
(2) The change in y-values over the change in x-values between two points using a formula
(3) The slope of a line is equal to the tangent of the angle of inclination
(4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses trigonometry concepts related to 3D shapes and solving problems involving angles of elevation. Specifically:
- When doing 3D trigonometry, it is often useful to redraw shapes in 2D to analyze them.
- A worked example problem is shown to find the distance and bearing between a life raft (David's position) and a search vessel (Anna's position) based on angles of elevation they each observe of a mountain peak.
- Applying trigonometric relationships involving angles and the mountain's known height, the distance between David and Anna is calculated to be 2799 meters, and the bearing of David from Anna is calculated to be 249°51'.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example problem demonstrating these steps, eliminating variables through multiplication and addition of the equations until a single variable remains that can be solved for. The document notes that simultaneous equations with the same number of variables and equations can always be solved using this method.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document discusses trigonometric functions, arcs, sectors, and related concepts. It defines:
- 360° = 2π radians
- The circumference of a circle is given by C = 2πr
- The area of a circle is given by A = πr^2
- The length of an arc is given by l = rθ
- The area of a sector is given by A = (1/2)r^2θ
It provides an example calculating the length of an arc and area of a sector for a circle with radius 5cm and central angle of 45°.
The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its direction. It is solved as the number if it is positive, and the opposite of the number if it is negative. Examples of evaluating absolute value expressions are provided. The document also discusses solving absolute value equations and inequalities, providing examples of setting up and solving different absolute value equations for the variable.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses coordinate geometry concepts including the distance formula and midpoint formula. It explains that the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint. It also discusses dividing intervals, noting that for a level 2 math exam it is restricted to midpoint divisions in a 1:1 ratio, while an extension 1 exam can involve any ratio for internal or external divisions. Examples are provided to illustrate the concepts.
The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope:
(1) The rise over the run between two points (vertical change over horizontal change)
(2) The change in y-values over the change in x-values between two points using a formula
(3) The slope of a line is equal to the tangent of the angle of inclination
(4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. These solutions form a regular n-sided polygon with vertices on the unit circle when placed on an Argand diagram. As an example, it shows that the solutions to z5 = 1 are the fifth roots of unity located at angles that are integer multiples of 2π/5 around the unit circle. It then proves that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4 and ω5 are also roots. Finally, it proves that 1 + ω + ω2 + ω3 + ω4 = 0.