6.5 – Inverse Trig Functions

Review/Warm Up 1) Can you think of an angle ϴ, in radians, such that sin(ϴ) = 1? 2) Can you think of an angle ϴ, in radians, such that cos(ϴ) = -√3/2 3) From precalculus, do you remember how to solve for the inverse function if y = 2x 3 + 1? 4) How can you verify whether two functions are inverses of one another? Use the inverse you found for the function above. 5) Say you know all three sides from a right triangle. Can you think of a way to determine the other missing degree angles?

Like other functions from precalculus, we may also define the inverse functions for trig functions In the case of trig function, why would the inverse be useful?

Say you know sin(ϴ) = 0.35 – Do we know an angle ϴ off the top of our heads that would give us this value? The inverse is there for us to now determine unknown angles

The Inverse Functions There are two ways to denote the inverse of the functions If y = sin(x), x = arcsin(y) OR If y = sin(x), x = sin -1 (y)

Similar applies to the others If y = cos(x), x = arccos(y) OR If y = cos(x), x = cos -1 (y) If y = tan(x), x = arctan(y) OR If y = tan(x), x = tan -1 (x)

Finding the inverse To find the inverse, or ϴ of each function, we generally will use our graphing calculator to help us Example. Evaluate arccos(0.3)

Example. Evaluate tan -1 (0.4) Example. Evaluate sin -1 (-1)

In the case of inverse trig functions, f -1 (f(x)) and f(f -1 (x)) is not necessarily = x Always evaluate trig functions as if using order of operations; inside of parenthesis first

Example. Evaluate arcsin(sin(3π/4)) – Do we get “x” back out?

Example. Evaluate cos(arctan(0.4))

Assignment Pg odd 40, 41