4 7 INVERSE TRIGONOMETRIC FUNCTIONS For an inverse

4. 7 INVERSE TRIGONOMETRIC FUNCTIONS

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

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

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

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

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

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

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,

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

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

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

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:

y = tan x

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

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

Using the calculator. • • • Be in radian mode Arctan(-15. 7896) Arcsin(. 3456) Arccos(-. 6897) Arcsin(1. 4535) Arccos(-2. 4534)

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