The nth roots of unity are the solutions to the equation zn = ±1.
When placed on an Argand diagram, the nth roots of unity form the vertices of a regular n-sided polygon on the unit circle.
For example, the fifth roots of unity are the solutions to z5 = 1, which are the five regularly spaced points forming a pentagon on the unit circle: 1, cis(2π/5), cis(-2π/5), cis(4π/5), cis(-4π/5).
The document provides instructions to sketch the graphs of various functions, including y=f(x), y=f(x)+f(x), y=[f(x)]^2, y=e^f(x), and y=log(f(x)). It also includes examples of similar questions from past papers. The sketches are to be drawn separately for each function with labeled axes and key features like stationary points marked.
X2 T03 06 chord of contact & properties [2011]Nigel Simmons
The document discusses drawing tangents from an external point T to a hyperbola. It is shown that the two points where the tangents intersect the hyperbola, P and Q, must lie on the chord of contact, whose equation is given. For a rectangular hyperbola with foci at (0,c) and (0,-c), the equation of the chord PQ is derived. It is also shown that the coordinates of the external point T are (2cpq/(p+q), 2c/(p+q)). Substituting these coordinates into the equation of PQ yields an equation involving only x and y.
The document discusses circle geometry and properties related to circles. It contains two main sections: (1) Converse circle theorems which describe properties of circles related to right triangles, angles subtended at points on a chord, and cyclic quadrilaterals. (2) The four centers of a triangle - the incentre, circumcentre, centroid, and orthocentre and their properties. It also discusses the interaction between geometry and trigonometry through a property relating side lengths and sine of angles of a triangle to the diameter of its circumcircle. An example problem demonstrates properties of angles and similar triangles related to a variable point on a circle.
The document discusses exponential growth and decay models. It shows that the differential equation for population growth or decay (dP/dt) is proportional to the current population (P). This leads to an exponential model of the form P=Ae^kt, where A is the initial population, k is the growth or decay constant, and t is time. Examples are given to illustrate calculating population sizes at given times and time periods for populations growing or decaying exponentially.
The document defines a geometric series as a sequence where each term is found by multiplying the previous term by a constant ratio. It provides the following key points:
- The constant ratio is called the common ratio (r)
- The general term of a geometric series is Tn = arn-1
- Examples are worked through to find the common ratio (r) and general term given various sequences
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, concyclic points, cyclic quadrilateral, concentric circles, and theorems about chords and arcs. Specifically, it states that (1) a perpendicular from the circle center to a chord bisects the chord, and the perpendicular bisector of a chord passes through the center, and (2) conversely, the line from the center to the midpoint of a chord is perpendicular to the chord.
The nth roots of unity are the solutions to the equation zn = ±1.
When placed on an Argand diagram, the nth roots of unity form the vertices of a regular n-sided polygon on the unit circle.
For example, the fifth roots of unity are the solutions to z5 = 1, which are the five regularly spaced points forming a pentagon on the unit circle: 1, cis(2π/5), cis(-2π/5), cis(4π/5), cis(-4π/5).
The document provides instructions to sketch the graphs of various functions, including y=f(x), y=f(x)+f(x), y=[f(x)]^2, y=e^f(x), and y=log(f(x)). It also includes examples of similar questions from past papers. The sketches are to be drawn separately for each function with labeled axes and key features like stationary points marked.
X2 T03 06 chord of contact & properties [2011]Nigel Simmons
The document discusses drawing tangents from an external point T to a hyperbola. It is shown that the two points where the tangents intersect the hyperbola, P and Q, must lie on the chord of contact, whose equation is given. For a rectangular hyperbola with foci at (0,c) and (0,-c), the equation of the chord PQ is derived. It is also shown that the coordinates of the external point T are (2cpq/(p+q), 2c/(p+q)). Substituting these coordinates into the equation of PQ yields an equation involving only x and y.
The document discusses circle geometry and properties related to circles. It contains two main sections: (1) Converse circle theorems which describe properties of circles related to right triangles, angles subtended at points on a chord, and cyclic quadrilaterals. (2) The four centers of a triangle - the incentre, circumcentre, centroid, and orthocentre and their properties. It also discusses the interaction between geometry and trigonometry through a property relating side lengths and sine of angles of a triangle to the diameter of its circumcircle. An example problem demonstrates properties of angles and similar triangles related to a variable point on a circle.
The document discusses exponential growth and decay models. It shows that the differential equation for population growth or decay (dP/dt) is proportional to the current population (P). This leads to an exponential model of the form P=Ae^kt, where A is the initial population, k is the growth or decay constant, and t is time. Examples are given to illustrate calculating population sizes at given times and time periods for populations growing or decaying exponentially.
The document defines a geometric series as a sequence where each term is found by multiplying the previous term by a constant ratio. It provides the following key points:
- The constant ratio is called the common ratio (r)
- The general term of a geometric series is Tn = arn-1
- Examples are worked through to find the common ratio (r) and general term given various sequences
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, concyclic points, cyclic quadrilateral, concentric circles, and theorems about chords and arcs. Specifically, it states that (1) a perpendicular from the circle center to a chord bisects the chord, and the perpendicular bisector of a chord passes through the center, and (2) conversely, the line from the center to the midpoint of a chord is perpendicular to the chord.
The document discusses how to sketch graphs based on their equations. It provides the following key points:
- Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0.
- Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0.
- Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1.
Once intercepts are found, curves can be sketched by
The document discusses parametric coordinates and compares them to Cartesian coordinates. It states that parametric coordinates describe a curve using two equations where points are defined by a single parameter value, while Cartesian coordinates use a single equation where points are defined by two number values. The document then provides an example of a parabola with the parametric coordinates x=2at, y=at^2 and discusses how to derive the Cartesian equation from the parametric form. It also identifies the focus of the parabola as (1/4,0) and calculates the parametric coordinates for the curve y=8x^2 as (t/16, t^2/32).
11X1 T07 03 angle between two lines (2011)Nigel Simmons
The document shows how to find the acute angle between two lines given their slopes. It provides examples that demonstrate using the formula tan(α) = (m1 - m2) / (1 + m1m2) to calculate the angle. For lines with slopes m1 and m2, this formula can be used to find the acute angle between them.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
The document discusses concavity, turning points, and inflection points of functions. It states that the second derivative measures concavity, known as change in slope. If the second derivative is positive, the curve is concave up, if negative it is concave down, and if zero there may be a point of inflection. Turning points occur at stationary points, where the first derivative is zero. If the second derivative is positive at a turning point, it is a minimum, and if negative, it is a maximum. A point of inflection is where the concavity changes, which can be seen by checking the sign of the second derivative on either side of the point.
The document discusses the conical pendulum. It shows the force diagram and derives the relationship that the depth of the pendulum below the pivot point (h) is equal to the square of the angular velocity (ω) divided by twice the acceleration due to gravity (g). This means that as the angular velocity increases, the depth or level of the bob decreases. An example is given where increasing the revolutions per minute from 60 to 90 would cause the bob to rise in level.
The document discusses several theorems about triangles and polygons:
- The angle sum of any triangle is 180 degrees. This is proven by constructing an auxiliary line and using properties of alternate and corresponding angles.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This is shown by constructing a parallel line and using properties of alternate and corresponding angles.
- The angle sum of any quadrilateral is 360 degrees. This is proven by splitting the quadrilateral into two triangles and using the fact that the angle sum of a triangle is 180 degrees.
- The angle sum of any pentagon is 540 degrees. The proof of this is started but not completed in the document.
11X1 T01 09 completing the square (2011)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
Here are the steps to solve the given exercise:
1) 3cei: c, e, i
2) 4deghi: d, e, g, h, i
3) 6acm: a, c, m
4) 7ac: a, c
5) 8: 8
6) 11: 11
The letters in each term are written out separately. The numbers are simply written as numbers.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
11 x1 t05 05 arrangements in a circle (2013)Nigel Simmons
The document discusses the difference between arranging objects in a line versus in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, in a circular arrangement there is no definite start or end, so the number of arrangements is n! rather than n times the factorial of the remaining objects. Examples are provided to illustrate calculating the number of possible arrangements in different circular seating scenarios.
The document discusses real numbers and their properties. It covers prime factorization, finding the highest common factor (HCF), finding the lowest common multiple (LCM), and divisibility tests. Prime factorization represents a number as the product of its prime factors. To find the HCF, write numbers as products of their prime factors and take the common factors. To find the LCM, write numbers as products of their prime factors and take the factors using the highest powers. Divisibility tests provide shortcuts to determine if a number is divisible by 2, 3, 4, 5, 6, or 7.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document discusses graphs of the form y = x^n where n is an integer greater than 1. It notes that the graphs can first be sketched by noticing that all points where the curve cuts the x-axis are stationary points, and the curve has the same sign as the given value of n for any x. The graphs are also affected by whether n is odd or even.
11X1 T06 06 line through pt of intersectionNigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
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 discusses how to sketch graphs based on their equations. It provides the following key points:
- Numbers on axes must be evenly spaced. The y-intercept occurs when x=0 and the x-intercept occurs when y=0.
- Common curves include straight lines, parabolas, cubics, and higher order polynomials. Parabolas have x-intercepts found by solving the equation for where it equals 0.
- Higher order polynomials become flatter at the base and steeper on the sides as the power increases. Hyperbolas can be defined by equations like y=1/x or xy=1.
Once intercepts are found, curves can be sketched by
The document discusses parametric coordinates and compares them to Cartesian coordinates. It states that parametric coordinates describe a curve using two equations where points are defined by a single parameter value, while Cartesian coordinates use a single equation where points are defined by two number values. The document then provides an example of a parabola with the parametric coordinates x=2at, y=at^2 and discusses how to derive the Cartesian equation from the parametric form. It also identifies the focus of the parabola as (1/4,0) and calculates the parametric coordinates for the curve y=8x^2 as (t/16, t^2/32).
11X1 T07 03 angle between two lines (2011)Nigel Simmons
The document shows how to find the acute angle between two lines given their slopes. It provides examples that demonstrate using the formula tan(α) = (m1 - m2) / (1 + m1m2) to calculate the angle. For lines with slopes m1 and m2, this formula can be used to find the acute angle between them.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
The document discusses concavity, turning points, and inflection points of functions. It states that the second derivative measures concavity, known as change in slope. If the second derivative is positive, the curve is concave up, if negative it is concave down, and if zero there may be a point of inflection. Turning points occur at stationary points, where the first derivative is zero. If the second derivative is positive at a turning point, it is a minimum, and if negative, it is a maximum. A point of inflection is where the concavity changes, which can be seen by checking the sign of the second derivative on either side of the point.
The document discusses the conical pendulum. It shows the force diagram and derives the relationship that the depth of the pendulum below the pivot point (h) is equal to the square of the angular velocity (ω) divided by twice the acceleration due to gravity (g). This means that as the angular velocity increases, the depth or level of the bob decreases. An example is given where increasing the revolutions per minute from 60 to 90 would cause the bob to rise in level.
The document discusses several theorems about triangles and polygons:
- The angle sum of any triangle is 180 degrees. This is proven by constructing an auxiliary line and using properties of alternate and corresponding angles.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles. This is shown by constructing a parallel line and using properties of alternate and corresponding angles.
- The angle sum of any quadrilateral is 360 degrees. This is proven by splitting the quadrilateral into two triangles and using the fact that the angle sum of a triangle is 180 degrees.
- The angle sum of any pentagon is 540 degrees. The proof of this is started but not completed in the document.
11X1 T01 09 completing the square (2011)Nigel Simmons
The document discusses the process of completing the square to solve quadratic equations. It shows how to complete the square for a general quadratic equation of the form ax2 + bx + c = 0 by grouping like terms and factorizing into a perfect square form. Examples are worked through, including solving the specific equations x2 + 6x - 7 = 0, x2 - 6x + 6 = 0.
The document uses mathematical induction to prove that 2n is greater than n^2 for all n greater than 4. It shows that the statement holds for n=5. It then assumes the statement is true for some integer k greater than 4, and proves that it must also be true for k+1. Therefore, by the principle of mathematical induction, the statement is true for all integers n greater than 4.
Here are the steps to solve the given exercise:
1) 3cei: c, e, i
2) 4deghi: d, e, g, h, i
3) 6acm: a, c, m
4) 7ac: a, c
5) 8: 8
6) 11: 11
The letters in each term are written out separately. The numbers are simply written as numbers.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
11 x1 t05 05 arrangements in a circle (2013)Nigel Simmons
The document discusses the difference between arranging objects in a line versus in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, in a circular arrangement there is no definite start or end, so the number of arrangements is n! rather than n times the factorial of the remaining objects. Examples are provided to illustrate calculating the number of possible arrangements in different circular seating scenarios.
The document discusses real numbers and their properties. It covers prime factorization, finding the highest common factor (HCF), finding the lowest common multiple (LCM), and divisibility tests. Prime factorization represents a number as the product of its prime factors. To find the HCF, write numbers as products of their prime factors and take the common factors. To find the LCM, write numbers as products of their prime factors and take the factors using the highest powers. Divisibility tests provide shortcuts to determine if a number is divisible by 2, 3, 4, 5, 6, or 7.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document discusses graphs of the form y = x^n where n is an integer greater than 1. It notes that the graphs can first be sketched by noticing that all points where the curve cuts the x-axis are stationary points, and the curve has the same sign as the given value of n for any x. The graphs are also affected by whether n is odd or even.
11X1 T06 06 line through pt of intersectionNigel Simmons
The document provides steps to find the equation of a line passing through the intersection of two other lines and a given point. It first finds the intersection point of the two initial lines as (-2,3). It then calculates the slope of the line between this point and the given point (1,2) as -3. Finally, it derives the equation of the line as 7x + 21y - 49 = 0.
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 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
6. Differentiating Trig
lim sin x 0 Area AOC Area Sector OAC
x 0
lim cos x 1
x 0
lim tan x 0
x 0
C
x A
O 1
7. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
lim cos x 1
x 0
lim tan x 0 B
x 0
C
tan x
x A
O 1
8. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B
x 0
C
tan x
x A
O 1
9. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
C
tan x
x A
O 1
10. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x A
O 1
11. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A
O 1
12. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A as x 0
O 1 x
1 1
sin x
13. Differentiating Trig
lim sin x 0
x 0
Area AOC Area Sector OAC Area AOB
1 1 2 1
lim cos x 1
x 0
11 sin x 1 x 1 tan x
2 2 2
lim tan x 0 B sin x x tan x
x 0
sin x x tan x
sin x sin x sin x
C
tan x
x 1
1
sin x cos x
x A as x 0
O 1 x
1 1
sin x
x
lim 1
x 0 sin x
16. 5x
e.g. i lim 1 ii lim
x
x 0 sin 5 x x 0 sin 3 x
17. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
18. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
1
3
19. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
3
20. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
21. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
22. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
23. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2
lim cos x sin x
x 0 h
h
24. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2 cos 2 2 cos 2 1
lim cos x sin x
x 0 h
h
25. 5x
e.g. i lim 1 ii lim
x
lim
1 3x
x 0 sin 5 x x 0 sin 3 x x 0 3 sin 3 x
y sin x 1
dy sin x h sin x 3
lim
dx x 0 h
sin x cosh cos x sinh sin x
lim
x 0 h
cosh 1
lim cos x
sinh
sin x
x 0 h h
h
2 cos 2 2
sinh 2 cos 2 2 cos 2 1
lim cos x sin x
x 0 h
h
2 h
sinh 2 sin
lim cos x sin x 2
x 0 h h
26. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
27. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
28. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
cos x 1 sin x 0
29. 2 h
sin
lim cos x
sinh
sin x h 2
x 0 h
2
h
sinh sin
h
lim cos x sin x h 2 sin
x 0 h
2
2
cos x 1 sin x 0
cos x
30. 2 h
sinh sin
2
lim cos x sin x h
x 0 h
2
h
sinh sin
h
2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0
cos x
31. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
32. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
33. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
34. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
cos x
dy
dx 2
35. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2
cos x
dy
dx 2
sin x
36. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2 y cos f x
cos x
dy
dx 2
sin x
37. 2 h
sin
lim cos x
sinh 2
sin x h
x 0 h
2
h
h
sin
sinh 2
lim cos x sin x h sin
x 0 h
2
2 y sin f x
cos x 1 sin x 0 dy
f x cos f x
dx
cos x
y cos x
y sin x
2 y cos f x
cos x
dy
f x sin f x
dy
dx 2 dx
sin x
40. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
41. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
42. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
43. y tan x
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
44. y tan x e.g. i y sin x 3
sin x
y
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
45. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x
dx cos 2 x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
46. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
cos 2 x sin 2 x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
47. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos 2 x sin 2 x 2 sec 2
dx x x
cos 2 x
1
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
48. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x
sec 2 x
y tan f x
dy
f x sec 2 f x
dx
49. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x dy sin x
sec x
2
dx cos x
y tan f x
dy
f x sec 2 f x
dx
50. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x
cos 2 x dy sin x
sec x
2
dx cos x
tan x
y tan f x
dy
f x sec 2 f x
dx
51. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x
sec x
2
dx cos x
tan x
y tan f x
dy
f x sec 2 f x
dx
52. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x
dy
f x sec 2 f x
dx
53. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x v y cos e x
dy
f x sec 2 f x
dx
54. y tan x e.g. i y sin x 3
dy
sin x 3 x 2 cos x 3
y dx
cos x
dy cos x cos x sin x sin x 1
ii y tan
dx cos 2 x x
dy 1 1
cos x sin x
2 2
2 sec 2
2 dx x x
cos x
1 iii y log cos x iv y tan 5 x
cos 2 x dy sin x dy
5 tan 4 x sec 2 x
sec 2 x dx cos x dx
tan x
y tan f x v y cos e x
dy
f x sec 2 f x dy
dx e x sin e x
dx