Section 4.7: Inverse Trig Functions

Recall: The main purpose of the inverse trig functions is to give you the angle measure which pairs with a specific sine, cosine or tangent ratio. ex) Determine the following values of θ :

a) b)

Each trigonometric statement has a matching inverse trigonometric statement:

TRIGONOMETRIC STATEMENT INVERSE TRIG. STATEMENT

Opp sinθ = θ = Hyp Opp tanθ = θ = Adj Adj cosθ = θ = Hyp

When you plug a positive ratio into any of the inverse trigonometric functions they will return a POSITIVE ACCUTE ANGLE

Negative ratios plugged into inverse trig functions are a little trickier. Where the Inverse Trig Functions Come From In order to create the inverse to any function f()x , it must be ______To create the inverse of:

Sine Inverse Sine Graph

Domain restriction: Domain: Range: Range:

Tangent Inverse Tangent Graph

Domain restriction: Domain: Range: Range:

Cosine Inverse Cosine Graph

Domain restriction: Domain: Range: Range: Pay careful attention to the ranges of the inverse trig functions! This three‐quarter unit circle summarizes the ranges of the inverse trig functions.

ex) Evaluate the following:

−1 1 sin (2 ) = ______(ask yourself ‘what angle from quadrant I has a 1 sine ratio of 2 ?’)

−1 1 sin (− 2 )= ______(ask yourself ‘what NEGATIVE angle from 1 quadrant IV has a sine ratio of ‐ 2 ?’)

−1 2 −1 2 cos (2 ) = ______cos (− 2 ) = ______

−1 −1 tan ( 3) = ______tan (− 3) = ______

ex) Compositions with inverse trig functions: Some may require the use of a ‘bowtie’ triangle

Evaluate:

−1 −1 3 cos( tan (− 1)) tan( sin (2 ))

−1 −1 4 sin( tan (− 5)) cos( sin (− 5 ))

ex) Using a right triangle diagram create an algebraic expression equivalent to these expressions. Assume that x > 0

−1 x −1 cot( cos (3 )) sec( tan (4x ))