Download y = arcsin x sin y = x. y = arccos x cos y = x.

Precalculus Notes
Lesson 4.7 Inverse Trigonometric Functions Part 1
Inverse Sine Function
Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse.
3π
2
π
−π
2
− 3π
2
2
Sin x has an inverse function on this restricted interval.
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y = arcsin x
The inverse sine function is defined by
if and only if
sin y = x.
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–π/2 , π/2].
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Example 1:
b. sin-1
a. arcsin
Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse.
− 3π
2
3π
2
π
−π
2
2
Cos x has an inverse function on this restricted interval.
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The inverse cosine function is defined by
y = arccos x
if and only if
cos y = x.
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , π].
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Example 2:
a. arccos
b. cos-1
Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse.
The inverse tangent function is defined by
π
y = arctan x
if and only if
tan y = x.
The domain of y = arctan x is (
.
The range of y = arctan x is [–π/2 , π/2].
2
− 3π
2
3π
2
−π
2
Tan x has an inverse function on this interval.
Example 3:
a. arctan
b. tan-1
Example 4: Use a calculator to approximate the value of each expression.
a. cos–1 0.75
b. arcsin 0.19
c. arctan 1.32
d. sin-1 (-1.1)
Example 5: Use an inverse function to write θ as a function of x.
Example 6:
A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters
from the launch pad. Let θ be the angle of elevation to the shuttle and let s be the height of the shuttle.
a. Write θ as a function of s.
b. Find θ if s = 300 meters.
Homework: Page 347 #5-19 odd, 43-47 odd, 105, 108, 109, 110, 111
c. Find θ if s = 1200 meters.