Inverse Trig Functions

Recall We know that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test.

Sine Wave From looking at a sine wave, it is obvious that it does not pass the Horizontal Line Test.

In order to pass the Horizontal Line Test (so that sin x has an inverse that is a function), we must restrict the domain. We restrict it to

Quadrant IV is Quadrant I is Answers must be in one of those two quadrants or the answer doesn’t exist.

How do we draw inverse functions? Switch the x’s and y’s! Switching the x’s and y’s also means switching the axis!

Sine Wave Domain/range of restricted wave? Domain/range of inverse?

Inverse Notation y = arcsin x or y = sin -1 x Both mean the same thing. They mean that you’re looking for the angle (y) where sin y = x.

Evaluating Inverse Functions Find the exact value of: Arcsin ½ –T–This means at what angle is the sin = ½ ? –π–π/6 –5–5π/6 has the same answer, but falls in QIII, so it is not correct.

Calculator When looking for an inverse answer on the calculator, use the 2 nd key first, then hit sin, cos, or tan. When looking for an angle always hit the 2 nd key first. Last example: Degree mode, 2 nd, sin,.5 = 30.

Evaluating Inverse Functions Find the value of: sin -1 2 –T–This means at what angle is the sin = 2 ? –W–What does your calculator read? Why? –2–2 falls outside the range of a sine wave and outside the domain of the inverse sine wave

Cosine Wave

We must restrict the domain Now the inverse

Quadrant I is Quadrant II is Answers must be in one of those two quadrants or the answer doesn’t exist.

Tangent Wave

We must restrict the domain Now the inverse

Graphing Utility: Graphs of Inverse Functions Graphing Utility: Graph the following inverse functions. a. y = arcsin x b. y = arccos x c. y = arctan x – ––  – 2 –– –3 3  –– Set calculator to radian mode.

Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. a. cos – b. arcsin 0.19 c. arctan 1.32d. arcsin 2.5 Set calculator to radian mode.

Composition of Functions Find the exact value of Where is the sine = Replace the parenthesis in the original problem with that answer Now solve

Example Find the exact value of The sine angles must be in QI or QIV, so we must use the reference angle

Find tan(arctan(-5)) -5 Find If the words are the same and the inverse function is inside the parenthesis, the answer is already given!

Find the exact value of Steps: Draw a triangle using only the info inside the parentheses. Now use your x, y, r’s to answer the outside term 2 3

Last Example Find the exact value of Cos is negative in QII and III, but the inverse is restricted to QII

You Do Find the exact value of