1. 4.7 INVERSE TRIGONOMETRIC FUNCTIONS

2. For an inverse to exist the function MUST be one- to - one A function is one-to- one if for every x there is exactly one y and for every y there is exactly one x. So If x and/or y is raised to an even power then the inverse does not exist unless the domain is restricted.

3. The equation y = x 2 does not have an inverse because two different x values will produce the same y- value. i.e. x = 2 and x = -2 will produce y = 4. The horizontal line test fails. In order to restrict the domain, a basic knowledge of the shape of the graph is crucial. This is a parabola with (0,0) as the vertex. Restrict the domain to the interval [0,infinity) to make it one-to-one.

4. Now let’s look at the trig functions y = sin x y = cos x y = tan x

5. For the graph of y = sin x, the Domain is (-∞, ∞) the Range is [-1, 1] Not a 1-1 function So it currently does not have an inverse

6. However we can restrict the domain to [- p/2, p/2] Note the range will remain [-1, 1] Now it’s 1-1!

7. y = sinx The inverse of sinx or Is denoted as arcsinx

8. On the unit circle: For the inverse sine function with angles only from - p/2 to p/2 our answers will only be in either quadrant 1 for positive values and quadrant 4 for negative values. Find the exact value, if possible,

9. y = cos x is not one to one, so its domain will also need to be restricted.

10. y = cos x is not one to one, so its domain will also need to be restricted.

11. On this interval, [0, p ] the cosine function is one-to- one and we can now define the inverse cosine function. y = arccos x or y = cos -1 x y = arccos x y = cos x

12. On the unit circle, inverse cosine will only exist in quadrant 1 if the value is positive and quadrant 2 if the value is negative. Find the exact value for:

13. y = tan x

14. Remember that tangent is undefined at - p/2 and p /2 y = tanx y = arctanx

15. Remember that tangent is undefined at - p/2 and p /2 Find the exact value

16. Using the calculator. Be in radian mode Arctan(-15.7896) Arcsin(.3456) Arccos(-.6897) Arcsin(1.4535) Arccos(-2.4534)

17. H Dub 4-7 Page 349 #1-16all, 49-67odd