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 = 2x3 + 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. 527• 5-33odd• 40, 41