Section 8.1: The Inverse Sine, Cosine, and Tangent Functions

Transcription

1 Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function to only on cycle; i.e., to the interval [ π, ] π, the the function is one-to-one and so it does have an inverse. Def: The inverse sine, also called the arcsine, is the function y = sin 1 x = arcsin x, which is the inverse of the function x = sin y. The domain of the inverse sine is 1 x 1 and the range is π y π. The graph of y = sin 1 x looks like: Since sin x and sin 1 x are inverses of each other, we have the following relationships: 1. sin 1 (sin x) = x, provided that π x π.. sin ( sin 1 x ) = x, provided that 1 x 1. In the first equation, if x is not between π and π, then you first need to figure out which quadrant x is in. If x is in quadrants I or IV, then change x to its coterminal angle which is between π and π. If x is in quadrant II, change x for its reference angle. If x is in quadrant III, change x to the angle in quadrant IV which has the same reference angle as x. In the second equation, if x is not between 1 and 1, then the composition is undefined. Def: The inverse cosine, also called the arccosine, is the function y = cos 1 x = arccos x, which is the inverse of the function x = cos y. The domain of the 1

2 inverse cosine is 1 x 1 and the range is 0 y π. The graph of y = cos 1 x looks like: Since cos x and cos 1 x are inverses of each other, we have the following relationships: 1. cos 1 (cos x) = x, provided that 0 x π.. cos (cos 1 x) = x, provided that 1 x 1. In the first equation, if x is not between 0 and π, then you first need to figure out which quadrant x is in. If x is in quadrants I or II, then change x to its coterminal angle which is between 0 and π. If x is in quadrant III, change x to the angle in quadrant II which has the same reference angle as x. If x is in quadrant IV, then change x for its reference angle. In the second equation, if x is not between 1 and 1, then the composition is undefined. Def: The inverse tangent, also called the arctangent, is the function y = tan 1 x = arctan x, which is the inverse of the function x = tan y. The domain of the inverse tangent is < x < and the range is π < y < π. The graph of y = tan 1 x looks like:

3 Since tan x and tan 1 x are inverses of each other, we have the following relationships: 1. tan 1 (tan x) = x, provided that π < x < π.. tan (tan 1 x) = x, provided that < x <. In the first equation, if x is not between π and π, then you first need to figure out which quadrant x is in. If x is in quadrants I or IV, then change x to its coterminal angle which is between π and π. If x is in quadrant II then change x to the angle in quadrant IV which has the same reference angle as x. If x is in quadrant III, then change x for its reference angle. ex. Find the exact value of each expression. ( ) (a) cos 1 (b) tan 1 ( 3 ) ex. Find the exact value, if any, of each expression. (a) sin [ 1 sin ( )] 3π 5 3

4 (b) sin [ sin ( )] (c) cos 1 [ cos ( 3π 4 )] (d) cos [cos 1 (π)] (e) tan [ 1 tan ( )] 11π 5 4

5 Section 8.: The inverse Trigonometric Functions (Continued) Def: The inverse secant, also called the arcsecant, is the function y = sec 1 x = arcsec x, which is the inverse of the function x = sec y. The domain of the inverse secant is (, 1] [1, ) and the range is [ 0, π ) ( π, π]. Def: The inverse cosecant, also called the arccosecant, is the function y = csc 1 x = arccsc x, which is the inverse of the function x = csc y. The domain of the inverse cosecant is (, 1] [1, ) and the range is [ π, 0) ( 0, π ]. Def: The inverse cotangent, also called the arccotangent, is the function y = cot 1 x = arccot x, which is the inverse of the function x = tan y. The domain of the inverse tangent is < x < and the range is 0 < y < π. Note: The inverse of a trig function is asking what angle in the domain would be needed to give the trig value the given value. So to find the exact value of a trig expression involving a trig function composed with an inverse trig function which are not inverses of each other, use the inverse trig function to draw a right triangle and use the triangle to solve the problem. ex. Find the exact value of each expression. (a) tan [ cos ( )] (b) sec [ cos 1 ( 3 4 )] 1

6 (c) sin 1 ( cos 3π 4 ) (d) cot ( csc 1 10 ) ex. Write each trigonometric expression as an algebraic expression in u. (a) cos ( sin 1 u ) (b) tan (csc 1 u)

7 Section 8.3 (Previously Section 8.7 & 8.8): Trigonometric Equations Recall that the period of sin x, cos x, csc x, & sec x is π and the period of tan x & cot x is π. Thus, θ (Degrees) sin (θ n) = sin θ cos (θ n) = cos θ tan (θ n) = tan θ csc (θ n) = csc θ sec (θ n) = sec θ cot (θ n) = cot θ θ (Radians) sin (θ + πn) = sin θ cos (θ + πn) = cos θ tan (θ + πn) = tan θ csc (θ + πn) = csc θ sec (θ + πn) = sec θ cot (θ + πn) = cot θ ex. Solve each equation on the interval 0 θ < π. (a) sin (θ) + 1 = 0 (b) sec θ = 4 1

8 (c) 4 sin θ 3 = 0 (d) cos ( θ ) π 3 4 = 1 ex. Give a general formula for all the solutions. List six solutions. (a) cos θ = 1 (b) cot θ = 1 (c) sin (θ) = 1 ex. Solve each equation on the interval 0 θ < π. (a) sin θ 3 sin θ + 1 = 0

9 (b) 8 1 sin θ = 4 cos θ (c) cos θ + cos (θ) = 0 (d) sin θ 3 cos θ = 3

10 Section 8.4 (Previously Section 8.3): Trigonometric Identities ex. Establish each identity. (a) tan θ cot θ sin θ = cos θ (b) cos θ cos θ sin θ = 1 1 tan θ 1

11 (c) 1 sin θ 1+cos θ = cos θ (d) csc θ sin θ = cos θ cot θ

12 Section 8.5 (Previously Section 8.4): Sum and Difference Formulas Theorem (Sum and Difference Formulas) 1. sin (x + y) = sin x cos y + cos x sin y. sin (x y) = sin x cos y cos x sin y 3. cos (x + y) = cos x cos y sin x sin y 4. cos (x y) = cos x cos y + sin x sin y 5. tan (x + y) = tanx+tan y 6. tan (x y) = 1 tan x tan y tan x tan y 1+tan x tan y ex. Find the exact value of each expression. (a) cos 15 (b) tan 75 (c) sin 165 (d) sec 105 (e) csc ( ) 11π 1 1

13 (f) cot ( ) 5π 1 ex. Find the exact value of (a) sin (x + y), (b) cos (x + y), (c) tan (x y) given that sin x = 3 5, π < x < 3π ; cos y = 1 13, 3π < y < π ex. Establish each identity. (a) sin (π + θ) = sin θ

14 (b) sin (x y) sin x cos y = 1 cot x tan y ex. Find the exact value of each expression. (a) cos ( sin 1 3 ) 5 cos 1 1 (b) tan [ sin ( ] 1 ) 1 tan

15 Section 8.6 (Previously Section 8.5): Double-angle and Half-angle Formulas Theorem (Double-angle Formulas) 1. sin (θ) = sin θ cos θ. cos (θ) = cos θ sin θ 3. cos (θ) = 1 sin θ 4. cos (θ) = cos θ 1 5. tan (θ) = tan θ 1 tan θ Note: Formulas 1,, and 5 can be obtained from the Sum Formulas from the previous section by setting x = θ and y = θ. Formulas 3 and 4 can be obtained from formula by using the Pythagorean Identity sin θ+cos θ = 1. In formula 3, solve the Pythagorean Identity for cos θ and plugging it into formula. In formula 4, solve the Pythagorean Identity for sin θ and plugging it into formula. From the Double-angle formulas, we can get formulas for the square of the trig functions. 1. sin θ =. cos θ = 3. tan θ = 1 cos (θ) 1+cos (θ) 1 cos (θ) 1+cos (θ) In the previous set of formulas for the square of the trig functions, if we replace each θ by φ, we get the following formulas: 1. sin φ = 1 cos φ. cos φ = 1+cos φ 3. tan φ = 1 cos φ 1+cos φ Theorem (Half-angle Formulas) 1. sin θ = ± 1 cos θ. cos θ = ± 1+cos θ 3. tan θ = ± 1 cos θ 1+cos θ 4. tan θ = 1 cos θ sin θ 5. tan θ = sin θ 1+cos θ where the + or sign is determined by the quadrant in which the angle θ lies in. 1

16 ex. Find (a) cos (θ), (b) sin θ given that sin θ = 3 5, π < θ < 3π ex. Find the exact value of each expression. (a) cos 15 (b) tan π 8 ex. Establish each identity. (a) sin (θ) cos (θ) = sin (4θ)

17 (b) sin (3θ) = 3 sin θ 4 sin 3 θ ex. Find the exact value of each expression. (a) sin ( ) 1 cos (b) tan ( ) sin

18 Section 8.7 (Previously Section 8.6): Product-to-Sum and Sum-to-Product Formulas Theorem (Product-to-Sum Formulas) 1. sin x sin y = 1 [cos (x y) cos (x + y)]. cos x cos y = 1 [cos (x y) + cos (x + y)] 3. sin x cos y = 1 [sin (x + y) + sin (x y)] Theorem (Sum-to-Product Formulas): 1. sin x + sin y = sin x+y cos x y. sin x sin y = sin x y cos x+y 3. cos x + cos y = cos x+y cos x y 4. cos x cos y = sin x+y sin x y ex. Express each product as a sum containing only sines or only cosines. (a) sin (3θ) sin (4θ) (b) cos (3θ) cos (θ) (c) sin ( ) ( θ cos 3θ ) ex. Express each sum or difference as a product of sines and/or cosines. (a) sin θ + sin (4θ) (b) cos (5θ) + cos θ 1

19 (c) cos ( ) ( θ cos 5θ ) ex. Establish each identity. (a) sin (θ)+sin (4θ) cos (θ)+cos (4θ) = tan (3θ) (b) sin θ [sin (3θ) + sin (5θ)] = cos θ [cos (3θ) cos (5θ)]