The document discusses travel graphs showing the height of a bouncing ball over time. It provides the equation to model the ball's height and uses it to:
1) Find the ball's height after 1 second (35m) and the time it reaches this height again (7 seconds)
2) Calculate the average velocity during the 1st second (35m/s) and 5th second (-5m/s)
3) Determine that the average velocity over the ball's 8 seconds in the air is 0.
12 x1 t04 07 approximations to roots (2013)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
The document discusses exponential growth and decay models. It shows that the rate of change of population (dP/dt) is proportional to the existing population (P), with dP/dt = kP. This leads to the differential equation P = Ae^kt, where A is the initial population and k is the growth or decay constant. Examples are given to show how to use the model to determine population sizes at different times, and to calculate growth or decay rates from population data.
12 x1 t04 03 further growth & decay (2012)Nigel Simmons
The document discusses equations to model growth and decay over time by accounting for limiting conditions. It presents an equation for population change over time and its solution. It then gives an example using Newton's law of cooling, showing that an equation in the form T = A + Ce^kt satisfies the law, where T is temperature, A is the constant outside temperature, and C and k are constants. It then solves this equation to find that given information about a room's temperature dropping over half an hour, the temperature will reach 10 degrees Celsius after approximately 2 hours.
The document discusses travel graphs showing the height of a bouncing ball over time. It provides the equation to model the ball's height and uses it to:
1) Find the ball's height after 1 second (35m) and the time it reaches this height again (7 seconds)
2) Calculate the average velocity during the 1st second (35m/s) and 5th second (-5m/s)
3) Determine that the average velocity over the ball's 8 seconds in the air is 0.
12 x1 t04 07 approximations to roots (2013)Nigel Simmons
The document describes the method of halving intervals to find approximations of roots. It begins by stating if a function f(x) is continuous on an interval [a,b] where f(a) and f(b) have opposite signs, then there exists a root between a and b. It then works through an example of finding the root of x4 + 2x - 19 = 0 between 1 and 3 by repeatedly halving intervals and evaluating the function at the midpoint until reaching an approximation of 1.96 to two decimal places.
12 x1 t04 06 integrating functions of time (2012)Nigel Simmons
The document discusses integrating functions of time to determine changes in displacement, distance, velocity, and speed. It explains that the integral of position over time equals displacement, while subtracting integrals of position over different time intervals equals distance. Similarly, the integral of velocity over time equals speed, while the integral of acceleration over time equals velocity. Graphs of functions and their derivatives are also presented, showing the relationships between integration and differentiation.
The document discusses exponential growth and decay models. It shows that the rate of change of population (dP/dt) is proportional to the existing population (P), with dP/dt = kP. This leads to the differential equation P = Ae^kt, where A is the initial population and k is the growth or decay constant. Examples are given to show how to use the model to determine population sizes at different times, and to calculate growth or decay rates from population data.
12 x1 t04 03 further growth & decay (2012)Nigel Simmons
The document discusses equations to model growth and decay over time by accounting for limiting conditions. It presents an equation for population change over time and its solution. It then gives an example using Newton's law of cooling, showing that an equation in the form T = A + Ce^kt satisfies the law, where T is temperature, A is the constant outside temperature, and C and k are constants. It then solves this equation to find that given information about a room's temperature dropping over half an hour, the temperature will reach 10 degrees Celsius after approximately 2 hours.
The document discusses calculating rates of change for variables in terms of other variables. It provides two examples:
1) A spherical balloon deflating at a constant radius decrease rate of 10 mm/s. When the radius is 100 mm, the volume decrease rate is calculated to be 400000π mm3/s.
2) A spherical bubble expanding with a constant volume increase rate of 70 mm3/s. When the radius is 10 mm, the surface area increase rate is calculated to be 140 mm2/s.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and determining values like speed, acceleration, and position at different points in time.
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(
11 x1 t05 04 point slope formula (2013)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) and parallel to the given line (3x + 4y - 5 = 0).
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
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 several triangle and polygon theorems:
- The sum of the interior angles of any triangle is 180 degrees.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- The sum of the interior angles of any quadrilateral is 360 degrees.
- The sum of the interior angles of any pentagon is 540 degrees.
Proofs of these theorems are presented using angle properties, parallel lines, and adding individual triangle angle sums.
11 x1 t07 04 quadrilateral family (2012)Nigel Simmons
The document presents a diagram showing the hierarchical classification of quadrilaterals. It begins with quadrilateral as the parent shape and branches down to more specific shapes such as squares, rectangles, rhombi, parallelograms, trapezoids, and kites. The text explains that quadrilaterals can be classified based on their properties, and moving down the hierarchy, the shapes become more specialized. It also notes that any property of the parent shapes also applies to the child shapes.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. There are four main tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of applying the SAS test to prove corresponding sides and angles of two triangles are congruent. Additionally, it defines different types of triangles such as isosceles and equilateral triangles based on their side lengths and angle measures.
The document discusses key concepts in Euclidean geometry including definitions, notation, terminology, naming conventions, angle theorems, and constructing proofs. It defines parallel lines, perpendicular lines, congruent lines, and similar lines using notation symbols. It provides examples of naming angles, polygons, and parallel lines in a consistent cyclic order. It also outlines the steps and logical structure required to properly construct geometric proofs.
11 x1 t05 06 line through pt of intersection (2013)Nigel 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.
The document discusses properties of transversals and ratios of intercepts formed when a transversal cuts parallel lines. It includes examples of calculating lengths of intercepts using given ratios and applying the ratio property that the ratio of corresponding intercepts is equal to the ratio of distances from the transversal to the parallel lines.
11 x1 t05 01 division of an interval (2013)Nigel Simmons
The document discusses division of intervals in coordinate geometry. It states that for a 2 unit math exam, interval division questions are restricted to finding the midpoint, which divides an interval in a 1:1 ratio. For Extension 1 exams, intervals can be divided in any ratio internally or externally. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint, while the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
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 calculating rates of change for variables in terms of other variables. It provides two examples:
1) A spherical balloon deflating at a constant radius decrease rate of 10 mm/s. When the radius is 100 mm, the volume decrease rate is calculated to be 400000π mm3/s.
2) A spherical bubble expanding with a constant volume increase rate of 70 mm3/s. When the radius is 10 mm, the surface area increase rate is calculated to be 140 mm2/s.
The document discusses graphing sine curve functions. It explains that a basic sine curve has the equation y = sin(x) with a period of 2π units. More generally, a sine curve has the equation y = a sin(bx + c) where the period is 2π/b units, the amplitude is a units, and c shifts the curve left or right. It provides an example of y = 5 sin(9x - π/2) with a period of 2π/9 units, amplitude of 5 units, and a right shift of π/18.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and determining values like speed, acceleration, and position at different points in time.
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(
11 x1 t05 04 point slope formula (2013)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it contains:
1) The point slope formula: y - y1 = m(x - x1)
2) An example that finds the equation of the line passing through (-3,4) and (2,-6).
3) A second example that finds the equation (3x + 4y + 6 = 0) of the line passing through (2,-3) and parallel to the given line (3x + 4y - 5 = 0).
The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.
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 several triangle and polygon theorems:
- The sum of the interior angles of any triangle is 180 degrees.
- The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- The sum of the interior angles of any quadrilateral is 360 degrees.
- The sum of the interior angles of any pentagon is 540 degrees.
Proofs of these theorems are presented using angle properties, parallel lines, and adding individual triangle angle sums.
11 x1 t07 04 quadrilateral family (2012)Nigel Simmons
The document presents a diagram showing the hierarchical classification of quadrilaterals. It begins with quadrilateral as the parent shape and branches down to more specific shapes such as squares, rectangles, rhombi, parallelograms, trapezoids, and kites. The text explains that quadrilaterals can be classified based on their properties, and moving down the hierarchy, the shapes become more specialized. It also notes that any property of the parent shapes also applies to the child shapes.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. There are four main tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of applying the SAS test to prove corresponding sides and angles of two triangles are congruent. Additionally, it defines different types of triangles such as isosceles and equilateral triangles based on their side lengths and angle measures.
The document discusses key concepts in Euclidean geometry including definitions, notation, terminology, naming conventions, angle theorems, and constructing proofs. It defines parallel lines, perpendicular lines, congruent lines, and similar lines using notation symbols. It provides examples of naming angles, polygons, and parallel lines in a consistent cyclic order. It also outlines the steps and logical structure required to properly construct geometric proofs.
11 x1 t05 06 line through pt of intersection (2013)Nigel 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.
The document discusses properties of transversals and ratios of intercepts formed when a transversal cuts parallel lines. It includes examples of calculating lengths of intercepts using given ratios and applying the ratio property that the ratio of corresponding intercepts is equal to the ratio of distances from the transversal to the parallel lines.
11 x1 t05 01 division of an interval (2013)Nigel Simmons
The document discusses division of intervals in coordinate geometry. It states that for a 2 unit math exam, interval division questions are restricted to finding the midpoint, which divides an interval in a 1:1 ratio. For Extension 1 exams, intervals can be divided in any ratio internally or externally. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint, while the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
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
5. General Solutions of
Trig Equations
sin x
x0
sin 1 x sin 1 x
sin 1 x or sin 1 x
6. General Solutions of
Trig Equations
sin x
x0 x0
sin 1 x sin 1 x
sin 1 x or sin 1 x
7. General Solutions of
Trig Equations
sin x
x0 x0
sin 1 x sin 1 x
sin 1 x sin 1 x
sin 1 x or sin 1 x
8. General Solutions of
Trig Equations
sin x
x0 x0
sin 1 x sin 1 x
sin 1 x sin 1 x
sin 1 x or sin 1 x sin 1 x or 2 sin 1 x
9. General Solutions of
Trig Equations
sin x
x0 x0
sin 1 x sin 1 x
sin 1 x sin 1 x
sin 1 x or sin 1 x sin 1 x or 2 sin 1 x
sin 1 x or 2 sin 1 x
10. General Solutions of
Trig Equations
sin x
x0 x0
sin 1 x sin 1 x
sin 1 x sin 1 x
sin 1 x or sin 1 x sin 1 x or 2 sin 1 x
sin 1 x or 2 sin 1 x
sin x
k 1k sin 1 x where k is an integer
14. cos x
x0
cos 1 x
cos 1 x
cos 1 x or 2 cos 1 x
15. cos x
x0 x0
cos 1 x
cos 1 x
cos 1 x or 2 cos 1 x
16. cos x
x0 x0
cos 1 x cos 1 x
cos 1 x cos 1 x
cos 1 x or 2 cos 1 x
17. cos x
x0 x0
cos 1 x cos 1 x
cos 1 x cos 1 x
cos 1 x or 2 cos 1 x cos 1 x or cos 1 x
18. cos x
x0 x0
cos 1 x cos 1 x
cos 1 x cos 1 x
cos 1 x or 2 cos 1 x cos 1 x or cos 1 x
cos 1 x or cos 1 x
cos 1 x or 2 cos 1 x
19. cos x
x0 x0
cos 1 x cos 1 x
cos 1 x cos 1 x
cos 1 x or 2 cos 1 x cos 1 x or cos 1 x
cos 1 x or cos 1 x
cos 1 x or 2 cos 1 x
cos x
2k cos 1 x where k is an integer
23. tan x
x0
tan 1 x
tan 1 x
tan 1 x or tan 1 x
24. tan x
x0 x0
tan 1 x
tan 1 x
tan 1 x or tan 1 x
25. tan x
x0 x0
tan 1 x tan 1 x
tan 1 x tan 1 x
tan 1 x or tan 1 x
26. tan x
x0 x0
tan 1 x tan 1 x
tan 1 x tan 1 x
tan 1 x or tan 1 x tan 1 x or 2 tan 1 x
27. tan x
x0 x0
tan 1 x tan 1 x
tan 1 x tan 1 x
tan 1 x or tan 1 x tan 1 x or 2 tan 1 x
tan 1 x or 2 tan 1 x
28. tan x
x0 x0
tan 1 x tan 1 x
tan 1 x tan 1 x
tan 1 x or tan 1 x tan 1 x or 2 tan 1 x
tan 1 x or 2 tan 1 x
tan x
k tan 1 x where k is an integer
38. 1
iii tan
3
1 1
k tan
3
where k is an integer
39. 1
iii tan
3
1 1
k tan
3
where k is an integer
k
6
40. 1
iii tan
3
1 1
k tan
3
where k is an integer
k
6
If 0 2
,
6 6
7
,
6 6
41. 1
iii tan iv sin sin 5
3 7
1 1
k tan
3
where k is an integer
k
6
If 0 2
,
6 6
7
,
6 6
42. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
where k is an integer
k
6
If 0 2
,
6 6
7
,
6 6
43. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
2
where k is an integer k 1k
7
k where k is an integer
6
If 0 2
,
6 6
7
,
6 6
44. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
2
where k is an integer k 1k
7
k where k is an integer
6
v cos 2 x cos
If 0 2 9
,
6 6
7
,
6 6
45. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
2
where k is an integer k 1k
7
k where k is an integer
6
v cos 2 x cos
If 0 2 9
2 x 2k cos cos
1
, 9
6 6
7
,
6 6
46. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
2
where k is an integer k 1k
7
k where k is an integer
6
v cos 2 x cos
If 0 2 9
2 x 2k cos cos
1
, 9
6 6
2 x 2k
7 9
,
6 6
47. 1
iii tan iv sin sin 5
3 7
1 1 5
k tan k 1k sin 1 sin
3 7
2
where k is an integer k 1k
7
k where k is an integer
6
v cos 2 x cos
If 0 2 9
2 x 2k cos cos
1
, 9
6 6
2 x 2k
7 9
,
6 6 x k
18
where k is an integer