4.7Inverse Trig Functions
JMerrill, 2010
We know that for a function to have an inverse function, it must be one-to-one (it must pass the Horizontal Line Test).
Recall
From looking at a sine wave, it is obvious that it does not pass the Horizontal Line Test.
Sine Wave
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
The Unit Circle
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
Sine Wave
,2 2
Quadrant IV is Quadrant I is Answers must be in one of those two
quadrants or the answer doesn’t exist.
Sine Wave,0
2
0,
2
How do we draw inverse functions? Switch the x’s and y’s!
Sine Wave
Switching the x’s and y’s also means switching the axis!
Domain/range of restricted sine wave? Domain/range of inverse?
Sine Wave
: ,2 2
: 1,1
D
R
: 1,1
: ,2 2
D
R
y = arcsin x or y = sin-1 x
Both mean the same thing. They mean that you’re looking for the angle where sin y = x.
Inverse Notation
Find the exact value of: Arcsin ½
◦ This means at what angle is the sin = ½ ?◦ π/6◦ (5π/6 has the same answer, but falls in QIII, so it is
not correct)
Evaluating Inverse Functions
When looking for an inverse answer on the calculator, use the 2nd key first, then hit sin, cos, or tan.
When looking for an angle always hit the 2nd key first.
Last example: ◦ Degree mode, 2nd, sin, .5 = 30. ◦ Radian mode: 2nd, sin, .5 = .524 (which is pi/6)
Calculator
Find the value of: Sin-1 2
◦ This means at what angle is the sin = 2 ?◦ What does your calculator read? Why?
◦ 2 falls outside the domain of an inverse sine wave
Evaluating Inverse Functions
Cosine Wave
Domain and range of restricted wave? Domain and range of the inverse?
Cosine Wave
D: 0,R : 1,1
D: 1,1R : 0,
We must restrict the domain Now the inverse
Cosine Wave
Quadrant I is 0,2Quadrant I I is ,2
Tangent Wave
We must restrict the domain Now the inverse
Tangent Wave
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
–
–3 3
–
Set calculator to radian mode.
Graphing Utility: Approximate the value of each expression.
a. cos–1 0.75 b. arcsin 0.19
c. arctan 1.32 d. arcsin 2.5
Set calculator to radian mode.
Previously learned notation: ◦ fog(x) gof(x)
Composition of Functions
Find tan(arctan(-5))◦ -5 (the tangent and its inverse cancel each other
out!) Find arcsin(sin )
◦ 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
Find ◦ Outside the domain
Composition of Functions Using Inverse Properties
53
,2 2 5
3
53
3
cos(cos- )
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!)
Composition of Functions2tan arccos3
Example: 2Find the exact value of tan arccos .3
x
y
3
2
cos , x 2, 3 xSince rr
2 23 2 5
y 5tan x 2
Positive so draw in Q1)
Find the
You Try1 3cos sin 5
45
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