Section 8.1 The Inverse Sine, Cosine, and Tangent Functions You learned about inverse unctions in both college algebra and precalculus. The main characteristic o inverse unctions is that composing one within the other always equals x. In mathematical notation, this is written as ollows: 1 1 ( ) and ( ) x x x x. For instance, i ( x) x 7, then unctions: 1 x 7 ( x). Let s prove that these are inverse You may also remember that the domain (x-values) o ( x) is the same as the range (y-values) o 1 ( x), and vice versa. Thus, you can get points on one graph just by interchanging the x and y values rom the other. This results in graphs that are symmetric across the line y = x. The graphs o ( x) x 7 and 1 x 7 ( x) are shown in the igure to the right. Notice that they are mirror images across the line y = x. Notice also that ( x) contains the points (-, 1) and (-1, 4), and that 1 ( x) contains the points and. In order to have an inverse, a unction must be one-to-one, which means that its graph must pass the Horizontal Line Test. I a unction is not one-to-one, it is usually possibly to restrict its domain to make it one-to-one. Think o the graphs o the trig unctions, such as sine, cosine, and tangent. Are these unctions one-to-one? In order to have inverse trig unctions, we must restrict the domains o the unctions to make them one-to-one. We will do this as ollows: or sin( x ), restrict the domain to x, and or tan(x), restrict the domain to x (so we are only looking at sine and tangent or angles in Quadrants and ). For cos( x ), restrict the domain to 0 x (so we are only looking at cosine or angles in Quadrants and ). These restricted domains become the o the inverse trig unctions. Page 1
Now that we have restricted the domains o sine, cosine, and tangent, we are ready to deine their inverse unctions. sin(x) with its restricted domain cos(x) with its restricted domain tan(x) with its restricted domain Notice that the RANGE o the inverse trig unction is the same as the restricted DOMAIN o the original trig unction. FINDING THE EXACT VALUE OF INVERSE TRIGONOMETRIC FUNCTIONS To ind the answer to an inverse trig problem, you really just have to rearrange it. For instance, to ind the solution to sin 1, the question you really ask yoursel is this: Sine o what angle (where the angle has to be rom either quadrant or ) equals? Since sine is negative or angles in quadrant, we know our angle has to come rom quadrant. So what is the answer? What is the angle in quadrant whose sine equals Answer:? Page
Let s look at another one: cos. The question is this: Cosine o what angle (rom quadrant or ) equals 1 1 1? Since cosine is negative or angles in quadrant, we know our answer has to come rom that quadrant. So what is the angle in quadrant whose cosine equals Example: Find the exact value o each expression. a) sin -1 (0) (The answer has to be an angle between and.) 1? Answer: b) cos 1 (The answer has to be an angle rom quadrant or ; which one is it? ) 1 c) tan (The answer has to be an angle rom quadrant or ; which one is it? ) USING PROPERTIES OF INVERSE FUNCTIONS TO FIND EXACT VALUES OF COMPOSITE FUNCTIONS These three boxes illustrate a point we made 1 1 earlier: that ( ) and ( ) x x x x. In addition, they speciy the domain o each unction. Notice that the domain is dierent depending on whether the inverse unction is on the inside or the outside. I the inverse unction is on the outside, the answer has to lie in the previously-mentioned quadrants. Page
Example: Find the exact value o the composite unction. Notice that the inverse unction is on the outside. a) sin 1 sin 10 The angle is between and (as required by the deinition in the sine box on the 10 previous page). Thus, the answer is. b) cos 1 5 cos The angle 5 is not between 0 and. What quadrant is 5 in? So this angle actually lies in one o the allowable quadrants; we just need to rewrite it so that it is between 0 and. The answer is to write its reerence angle (remember that a reerence angle is the shortest distance between the given angle and the x-axis). Thus, we can re-write the problem as : by deinition, equals. cos 1 cos, which, c) 1 4 tan tan 5 The angle 4 is not between and (as required by the deinition in the tangent box 5 on the previous page). This angle is in Quadrant. In this quadrant, tangent is. In what other quadrant does tangent have this sign? Is that quadrant one o the ones that our 4 angle can be rom? So tan 5 = tan. Then we can rewrite the problem as tan 1 tan, which, by deinition, equals. d) tan 1 tan The angle is not between and. In act, the angle is in Quadrant. In this quadrant, tangent is. In what allowable quadrant does tangent have that sign? So tan would be the same as tangent o what angle? Thus, we can rewrite the problem as tan 1 tan, and then by deinition, this equals. Page 4
Example: Find the exact value o the composite unction. Notice that the inverse unction is on the inside. a) 1 coscos Since When the inverse cosine unction is on the inside, the x-value must be between -1 and 1. is in this interval, the answer is simply. 1 b) tantan When inverse tangent is on the inside, the x-value can be any real number x. 1 Thereore, since - is a real number, then tantan 1 c) sinsin =. When inverse sine is on the inside, the x-value must be between -1 and 1. Since - is NOT in this interval, we say the answer is "Not Deined". 1 d) cos cos 1.5 When inverse cosine is on the inside, the x-value must be between -1 and 1. Since 1.5 is NOT in this interval, the answer is. FINDING THE INVERSE FUNCTION OF A TRIGONOMETRIC FUNCTION We do this using the same steps we did in college algebra and precalculus. Step 1) Replace ( x) with y. Step ) Switch all y's and x's. Step ) Solve or y. Step 4) Replace y with 1 ( x). Example: Find the inverse unction and state the domain and range o and -1. a) ( x) tanx b) x x ( ) cos 1 Page 5
SOLVING EQUATIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS Example: Find the exact solution o each equation. a) cos 1 x b) 1 6sin ( x) c) 1 4 tan x Page 6