1. Inverse Trigonometric Functions
Digital Lesson
Inverse Trigonometric Functions

2. 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. y x
y = sin x
Sin x has an inverse function on this interval.
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Inverse Sine Function

3. The range of y = arcsin x is [–/2 , /2].
The inverse sine function is defined by
y = arcsin x if and only if sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–/2 , /2].
Example:
This is another way to write arcsin x.
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Inverse Sine Function

4. Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse. y x
y = cos x
Cos x has an inverse function on this interval.
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Inverse Cosine Function

5. Inverse Cosine Function
The inverse cosine function is defined by
y = arccos x if and only if cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , ].
Example:
This is another way to write arccos x.
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Inverse Cosine Function

6. Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse. y x
y = tan x
Tan x has an inverse function on this interval.
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Inverse Tangent Function

7. Inverse Tangent Function
The inverse tangent function is defined by
y = arctan x if and only if tan y = x.
Angle whose tangent is x
The domain of y = arctan x is
The range of y = arctan x is [–/2 , /2].
Example:
This is another way to write arctan x.
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Inverse Tangent Function

8. Example: Evaluating Composition of Functionsy 3 u 2
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Example: Evaluating Composition of Functions

9. Deriving Inverse Trig Fuctions
Tips to remember:
If it starts with ‘c’, make it negative!

10. Deriving Inverse Trig Fuctions

11. Deriving Inverse Trig Fuctions

12. Another way to write inverse functions

13. Spotlight Search…
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14. …Using inverse trig functions: