Notes 4.7 Inverse Trig Functions
Inverse sine function
Domain Inverse Sine Function
For a function to have an inverse function, it must pass the Horizontal Line Test
Range ­ Since the sine function does not pass the horizontal line test, the domain must be restricted to [­π/2, π /2].
• y = sin ­1 x can be thought of as the angle whose sine is x.
• note: y = sin ­1 x is the same thing as
y = arcsin x.
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Ex 1) Evaluate the following.
a) sin
­1
Other Inverse Trig Functions
(­1) = Graphs of the three inverse functions
b) arcsin (1/2) = c) sin
­1
(­√3 / 2) = Nov 5­8:02 AM
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Ex 3) Evaluate the following.
a) cos ­1 (­√3/2) = b) tan
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­1
(1) = Nov 5­8:12 AM
1
Ex 4) Use a calculator to evaluate the following.
•
sin ­1 (0.5524) = III. Compositions of Functions
• Remember:
f(f ­1 ( x)) = x and f
­1
(f ( x)) = x Inverse properties of trig functions:
1) If ­1 ≤ x ≤ 1 and –π/2 ≤ y ≤ π/2 then sin(sin ­1 x) = x and sin ­1 (sin x) = x
•
­1
tan (­3.254) =
2) If ­1 ≤ x ≤ 1 and 0 ≤ y ≤ π then cos(cos ­1 x) = x and cos
­1
(cos x) = x
3) If x is a real number and –π/2 ≤ y ≤ π/2 then tan(tan ­1 x) = x and tan ­1 (tan x) = x
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Ex 5) Evaluate the following.
Ex 6) Find the exact value.
a) sin(sin
­1 0.12) = • cos(sin
­1 (3/5)) =
• sin(tan
­1 (­1/2)) =
b) tan ­1 (tan 5π/6) =
c) cos(cos
­1 6) =
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Exit Problems
1) Find the exact value of the expression without using a calculator.
a) sin ­1 ½
b) tan ­1 ­√3
2) Find the exact value of the expression. (Hint: make a sketch of a right triangle.)
sin(tan ­1 4/3)
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2