# JMerrill, 2010. WWe know that for a function to have an inverse function, it must be one-to-one (it must pass the Horizontal Line Test).

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JMerrill, 2010

WWe know that for a function to have an inverse function, it must be one-to-one (it must pass the Horizontal Line Test).

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

 The Unit Circle in found in section 4.7.  We will use: ◦ Radians ◦ Exact answers (mostly) ◦ Quick board review of Unit Circle, quadrants on the wave, & converting to radian measure

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

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

DDomain/range of restricted sine wave? DDomain/range of inverse?

yy = arcsin x or y = sin -1 x BBoth mean the same thing. They mean that you’re looking for the angle where sin y = x.

FFind the exact value of: AArcsin ½ ◦T◦This means at what angle is the sin = ½ ? ◦π◦π/6 ◦(◦(5π/6 has the same answer, but falls in QIII, so it is not correct)

 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. ◦ Radian mode: 2 nd, sin,.5 =.524 (which is pi/6)

FFind the value of: S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 domain of an inverse sine wave

DDomain and range of restricted wave? DDomain and range of the inverse?

WWe must restrict the domain NNow the inverse

WWe must restrict the domain N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 –1.5 1.5 ––  –1.5 1.5 22 –– –3 3  –– Set calculator to radian mode.

Graphing Utility: Inverse Functions Graphing Utility: Approximate the value of each expression. a. cos – 1 0.75b. arcsin 0.19 c. arctan 1.32d. arcsin 2.5 Set calculator to radian mode.

 Previously learned notation: ◦ f o g(x)g o f(x)

FFind tan(arctan(-5)) ◦-◦-5 (the tangent and its inverse cancel each other out!) FFind arcsin(sin ) ◦T◦The domain of the sine function is. Since is outside that domain, we’ll just say that the answer is: is outside the domain (unless you remember coterminal angles and can tell me the actual answer is FFind ◦O◦Outside the domain

 Find the exact value of  Draw the graph using only the info inside the parentheses. (Easy way—completely ignore the fact that you have inverses!)

Example: x y 3 2 Positive so draw in Q1)

 Find the

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