5-2 Verifying Trigonometric Identities

Transcription

1 5- Verifying Trigonometric Identities Verify each identity. 1. (sec 1) cos = sin 3. sin sin 3 = sin cos 4 5. = cot 7. = cot 9. + tan = sec Page 1

2 5- Verifying Trigonometric Identities 7. = cot 9. + tan = sec =1 Page

3 5- Verifying Trigonometric Identities =1 13. (csc cot )(csc + cot )= = sec Page 3

4 5- Verifying Trigonometric Identities 15. = sec csc4 4 cot = cot FIREWORKS If a rocket is launched from ground level, the maximum height that it reaches is given by h =, where is the angle between the ground and the initial path of the rocket, v is the rocket s initial speed, and g is the acceleration due to gravity, 9.8 meters per second squared. a. Verify that =.. b. Suppose a second rocket is fired at an angle of 80 from the ground with an initial speed of 110 meters per second. Find the maximum height of the rocket. a. Page 4

5 5- Verifying Trigonometric Identities 19. FIREWORKS If a rocket is launched from ground level, the maximum height that it reaches is given by h = is the angle between the ground and the initial path of the rocket, v is the rocket s initial speed,, where and g is the acceleration due to gravity, 9.8 meters per second squared. a. Verify that =.. b. Suppose a second rocket is fired at an angle of 80 from the ground with an initial speed of 110 meters per second. Find the maximum height of the rocket. a. b. Evaluate the expression for v = 110 m,, and g = 9.8 m/s. The maximum height of the rocket is about meters. Verify each identity. 1. sin tan = tan sin Page 5

6 5- Verifying Trigonometric Identities The maximum height of the rocket is about meters. Verify each identity. 1. sin tan = tan sin 3. = cos + cot Page 6

7 5- Verifying Trigonometric Identities 3. = cos + cot 5. = Page 7

8 5- Verifying Trigonometric Identities 5. = 7. sec cos = tan sin Page 8

9 5- Verifying Trigonometric Identities 7. sec cos = tan sin 9. (csc cot ) = 31. = (sin + cos ) Page 9

10 5- Verifying Trigonometric Identities 31. = (sin + cos ) 33. PHOTOGRAPHY The amount of light passing through a polarization filter can be modeled using I = Im cos, where I is the amount of light passing through the filter, Im is the amount of light shined on the filter, and angle of rotation between the light source and the filter. Verify that Im cos = Im is the. GRAPHING CALCULATOR Test whether each equation is an identity by graphing. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. 35. sec x + tan x = Graph and then graph TheManual equation appears to be an esolutions - Powered by Cognero algebraically.. identity because the graphs of the related functions coincide. Verify this Page 10

11 5- Verifying Trigonometric Identities GRAPHING CALCULATOR Test whether each equation is an identity by graphing. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. 35. sec x + tan x = and then graph Graph. The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically. 37. = 1 sin x Graph Y1 = and then graph Y = 1 sin x. Page 11

12 5- Verifying Trigonometric Identities 37. = 1 sin x and then graph Y = 1 sin x. Graph Y1 = The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically. 39. cos x sin x = Graph Y1 = cos x sin x and then graph Y =. The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically. Page 1

13 5- Verifying Trigonometric Identities 39. cos x sin x = Graph Y1 = cos x sin x and then graph Y =. The equation appears to be an identity because the graphs of the related functions coincide. Verify this algebraically. Verify each identity. 41. = Page 13

14 5- Verifying Trigonometric Identities Verify each identity. 41. = 43. ln cot x + ln tan x cos x = ln cos x Verify each identity. 45. cos 4 = sin cos 4 1 Start with the right side of the identity sec tan = sec 6 tan Start with the right side of the identity. Page 14

15 5- Verifying Trigonometric Identities sec tan = sec 6 tan Start with the right side of the identity. 49. sec x csc x = sec x + csc x Start with the left side of the identity. HYPERBOLIC FUNCTIONS The hyperbolic trigonometric functions are defined in the following ways. sinh x = x x (e e ) esolutions Manual - Powered by x xcognero cosh x = (e + e ) Page 15

16 5- Verifying Trigonometric Identities HYPERBOLIC FUNCTIONS The hyperbolic trigonometric functions are defined in the following ways. x x sinh x = (e e ) cosh x = (e + e ) x x tanh x = csch x =, x 0,x 0 sech x = coth x = Verify each identity using the functions shown above. 51. cosh x sinh x = sech x = 1 tanh x GRAPHING CALCULATOR Graph each side of each equation. If the equation appears to be an identity, verify it algebraically. 55. =1 Page 16

17 5- Verifying Trigonometric Identities GRAPHING CALCULATOR Graph each side of each equation. If the equation appears to be an identity, verify it algebraically. 55. =1 Graph Y1 = and Y = 1. The graphs appear to be the same, so the equation appears to be an identity. Verify this algebraically. 57. (tan x + sec x)(1 sin x) = cos x Page 17

18 5- Verifying Trigonometric Identities 57. (tan x + sec x)(1 sin x) = cos x 59. MULTIPLE REPRESENTATIONS In this problem, you will investigate methods used to solve trigonometric equations. Consider 1 = sin x. a. NUMERICAL Isolate the trigonometric function in the equation so that sin x is the only expression on one side of the equation. b. GRAPHICAL Graph the left and right sides of the equation you found in part a on the same graph over [0, ). Locate any points of intersection and express the values in terms of radians. c. GEOMETRIC Use the unit circle to verify the answers you found in part b. d. GRAPHICAL Graph the left and right sides of the equation you found in part a on the same graph over < x <. Locate any points of intersection and express the values in terms of radians. e. VERBAL Make a conjecture as to the solutions of 1 = sin x. Explain your reasoning. b. Page 18

19 5- c. GEOMETRIC Use the unit circle to verify the answers you found in part b. d. GRAPHICAL Graph the left and right sides of the equation you found in part a on the same graph over <. Locate any points of intersection and express the values in terms of radians. e. VERBAL Trigonometric Make a conjecture asidentities to the solutions of 1 = sin x. Explain your reasoning. Verifying <x b. The graphs of y = sin x and y = intersect at and The graphs of y = sin x and y = intersect at, over [0, ). c. d., and over ( π, π). e. Sample answer: Since sine is a periodic function, the solutions of sin x = x= + nπ and x = are + n, where n is an integer. 61. CHALLENGE Verify that the area A of a triangle is given by A=, Page 19

20 e. Sample answer: Since sine is a periodic function, the solutions of sin x = are 5- xverifying = and x = where n is an integer. + nπ Trigonometric + n, Identities 61. CHALLENGE Verify that the area A of a triangle is given by A=, where a, b, and c represent the sides of the triangle and,, and are the respective opposite angles. Using the Law of Sines, A= ab sin A= a, so b =. Area of a triangle given SAS sin Substitution Multiply. A= α + β + γ = 180, so α = 180 (β + γ). A= A= = Sine Sum Identity A= A= sin 180 = 0, cos 180 = 1 Simplify. 63. OPEN ENDED Create identities for sec x and csc x in terms of two or more of the other basic trigonometric functions. Sample answers: tan x sin x + cos x = sec x and sin x + cot x cos x = csc x Page 0

21 A= = 5- AVerifying Trigonometric Identities sin 180 = 0, cos 180 = 1 Simplify. 63. OPEN ENDED Create identities for sec x and csc x in terms of two or more of the other basic trigonometric functions. Sample answers: tan x sin x + cos x = sec x and sin x + cot x cos x = csc x 65. Writing in Math Explain how you would verify a trigonometric identity in which both sides of the equation are equally complex. Sample answer: You could start on the left side of the identity and simplify it as much as possible. Then, you could move to the right side and simplify until it matches the left side. Simplify each expression. cot 67. tan 69. Page 1

22 5- Verifying Trigonometric Identities Locate the vertical asymptotes, and sketch the graph of each function. 73. y = tan x The graph of y = tan x is the graph of y = tan x compressed vertically. The period is or. Find the location of two consecutive vertical asymptotes. and Create a table listing the coordinates of key points for y = Function y = tan x y= tan x Vertical Asymptote Intermediate Point tan x for one period on x-intercept Intermediate Point. Vertical Asymptote x= (0, 0) x= x= (0, 0) x= Sketch the curve through the indicated key points for the function. Then repeat the pattern. Page

23 5- Verifying Trigonometric Identities Locate the vertical asymptotes, and sketch the graph of each function. 73. y = tan x tan x is the graph of y = tan x compressed vertically. The period is The graph of y = or. Find the location of two consecutive vertical asymptotes. and Create a table listing the coordinates of key points for y = Function Vertical Asymptote y = tan x y= tan x Intermediate Point tan x for one period on x-intercept Intermediate Point. Vertical Asymptote x= (0, 0) x= x= (0, 0) x= Sketch the curve through the indicated key points for the function. Then repeat the pattern. 75. y = sec 3x is the graph of y = sec x compressed vertically and horizontally. The period is The graph of or. Find the location of two vertical asymptotes. and Page 3

24 5- Verifying Trigonometric Identities 75. y = sec 3x is the graph of y = sec x compressed vertically and horizontally. The period is The graph of or. Find the location of two vertical asymptotes. and Create a table listing the coordinates of key points for Function y = sec x Vertical Asymptote x= Intermediate Point for one period on Vertical Asymptote Intermediate Point x=. Vertical Asymptote x= x= Sketch the curve through the indicated key points for the function. Then repeat the pattern. Write each degree measure in radians as a multiple of π and each radian measure in degrees Page 4

25 5- Verifying Trigonometric Identities Solve each inequality. 83. x + 3x 8 < 0 f(x) has real zeros at x = 7 and x = 4. Set up a sign chart. Substitute an x-value in each test interval into the polynomial to determine if f (x) is positive or negative at that point. The solutions of x + 3x 8 < 0 are x-values such that f (x) is negative. From the sign chart, you can see that the solution set is. 85. x + x 4 First, write x + x 4 as x + x 4 0. f(x) has real zeros at x = 6 and x = 4. Set up a sign chart. Substitute an x-value in each test interval into the polynomial to determine if f (x) is positive or negative at that point. Page 5

26 5- The solutions of x + 3x 8 < 0 are x-values such that f (x) is negative. From the sign chart, you can see that the solution set is Trigonometric. Verifying Identities 85. x + x 4 First, write x + x 4 as x + x 4 0. f(x) has real zeros at x = 6 and x = 4. Set up a sign chart. Substitute an x-value in each test interval into the polynomial to determine if f (x) is positive or negative at that point. The solutions of x + x 4 0 are x-values such that f (x) is positive or equal to 0. From the sign chart, you can see that the solution set is. 87. x 6x First, write x 6x +7 0 as x + 6x 7 0. f(x) has real zeros at x = 7 and x = 1. Set up a sign chart. Substitute an x-value in each test interval into the polynomial to determine if f (x) is positive or negative at that point. The solutions of x + 6x 7 0 are x-values such that f (x) is positive or equal to 0. From the sign chart, you can see that the solution set is. 89. SAT/ACT a, b, a, b, b, a, b, b, b, a, b, b, b, b, a, If the sequence continues in this manner, how many bs are there between the 44th and 47th appearances of the letter a? A 91 B 135 C 138 D 18 E 30 The number of bs after each a is the same as the number of a in the list (i.e., after the 44th a there are 44 bs). Between the 44th and 47th appearances of a the number of bs will be or 135. Therefore, the correct answer choice is B. 91. REVIEW Which of the following is not equivalent to cos, when 0 < θ <? Page 6

27 5- The number of bs after each a is the same as the number of a in the list (i.e., after the 44th a there are 44 bs). Between the 44th and 47th appearances of a the number of bs will be or 135. Verifying Trigonometric Identities Therefore, the correct answer choice is B. 91. REVIEW Which of the following is not equivalent to cos, when 0 < θ <? A B C cot D tan sin csc Therefore, the correct answer choice is D. Page 7