16 Inverse Trigonometric Functions
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1 6 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric Functions and the Periodicity Identities to Solve Trigonometric Equations. (Sections ) 6. The Problems with Inverse Trigonometric Functions Today we focus our attention on solving trigonometric equations. If we are going to have any hope of doing this in a systematic manner that does not involve a lot of guess and check, we need functions that can unravel or undo the trigonometric functions. In other words, we need inverse trigonometric function. Why is the notion of an inverse trigonometric function disturbing? In order to define inverse trigonometric functions, we first need to of the trigonometric functions so that the function is. There are multiple ways to of a trigonometric function so that we can define an inverse, but there are some restricted domains that are better than others. A good restricted domain has the following characteristics:
2 Even with these restrictions, we can see that there are multiple ways to define a good restricted domain. In the interest of consistency, uniformity, and transportability, most mathematicians and scientists have agreed on fairly standard restricted domains for the sine, cosine and tangent functions which will allow us to discuss the inverse sine (sin or arcsin) function, the inverse cosine (cos or arccos) function, and the inverse tangent (tan or arctan) function. 6. The Inverse Sine Function (sin or arcsin) The Restricted Domain of the Sine Function: [, ] y - x - Definition 6. (The Inverse Sine Function) For each v [,] there is a unique u [, ] such that sin(u) = v. We define sin (v) = u when sin(u) = v and u [, ]. NOTE: arcsin can replace sin in the previous definition. The input of the sin function is. The output of the sin function is. The input of the sin function is.
3 The output of the sin function is. Proposition 6. sin (sin(x)) = x if x [, sin(sin (x)) = x if x [,] ] Example 6. For each of the following, find an exact value for the expression or state that it is undefined. ( ) sin arcsin() ( ) arcsin ( ( )) sin sin sin ( sin ( )) 6 sin ( sin ( )) 5 6 sin(sin ()) sin (sin(.))
4 Example 6.4 Find the exact value of cos ( sin ( 5)). 6. The Inverse Cosine Function (cos or arccos) The Restricted Domain of the Cosine Function: [0, ] y x - Definition 6.5 (The Inverse Cosine Function) For each v [,] there is a unique u [0,] such that cos(u) = v. We define cos (v) = u when cos(u) = v and u [0,]. NOTE: arccos can replace cos in the previous definition. 4
5 Proposition 6.6 cos (cos(x)) = x if x [0,] cos(cos (x)) = x if x [,] Example 6.7 For each of the following, find an exact value for the expression or state that it is undefined. cos ( ) sin ( cos ( )) cos ( cos ( )) sin ( cos ( )) cos ( sin ( )) cos ( cos ( )) 5 6 Example 6.8 (Similar to Example 7 in Section 7.4 of your textbook.) Write cos(sin (x)) as an algebraic expression in x. 5
6 6.4 The Inverse Tangent Function (tan or arctan) The Restricted Domain of the Tangent Function: (, ) y x Definition 6.9 (The Inverse Tangent Function) For each v (, ) there is a unique u (, ) such that sin(u) = v. We define tan (v) = u when tan(u) = v and u (, ). NOTE: arctan can replace tan in the previous definition. Proposition 6.0 tan (tan(x)) = x if x (, tan(tan (x)) = x if x (, ) ) 6
7 Example 6. For each of the following, find an exact value for the expression or state that it is undefined. tan ( ) cos ( tan ( 8 5 ))) sin ( tan ( 8 5 ))) Example 6. Write an algebraic expression for sec (tan (x)) in terms of x. 7
8 6.5 Solving Some Trigonometric Equations The inverse trigonometric functions can be used to help you find a solution for some trigonometric equations. If you wish to find all solutions, you will also need to recall the periodicity identities. Recall that and sin( x) = sin(x) cos( x) = cos(x). Example 6. Find all solutions of the equations. Whenever possible, find exact solutions of the equation. sin(x) = cos(x)+0 = 0 8
9 sin(5x)+7 = 0 tan(x) = tan (x)+ = 5 9
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