Section 4.7 Inverse Trigonometric Functions

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Section 4.7Inverse Trigonometric Functions

• Objectives
• Evaluate inverse of sine, cosine, and tangent.
• Evaluate composite trig functions.

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Inverse Sine Wave
The inverse sine function is defined by y
arcsin x if and only if sin y x
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Domain and Range of Arcsin (Inverse Sine)

• Domain

Range
Notice
The domain of arcsine is the same as the range
of sine.
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Inverse Cosine Function

• Inverse Cosine (arccos)

Domain -1 x 1
Range 0 y
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Inverse Tangent function
Domain - lt x lt Range
lt y lt
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Trigonometric Function Restricted Domain Corresponding Range
Sin-1 x -1, 1 -p/2, p/2
Cos-1 x -1, 1 0, p
Tan-1 x (-8,8) -p/2, p/2
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Example 1

• Find the exact value.

• arcsin
• sin-1

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Example 2

• find the exact value.

• arccos
• tan-1 0
• cos-1 (-1)

0


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Example 3use calc. on f g

• a. Arcsin(1/2)
• b. Sin-1(v3/2)
• c. Sin-1(2)
• d. Arccos(v2/2)
• e. tan-1(v3/3)
• f. Arctan(-8.45)
• g. Sin-1(0.2447)

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Compositions of Trig Functions
The Domain (x) and Range (y) of each
function still applies.
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Example 4use triangles for d e

• a. Tan(arctan(-5))
• b. Arcsin(sin(5p/3))
• c. Cos(cos-1 p)
• d. Tan(arcsin(2/3))
• e. Cos(arctan(-3/4))

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Example 5using triangles
a.
b.
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Example 6

• Write an inverse trig function ? as a function of
  x.

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?
x5
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Example 7

• Write an algebraic function that is equivalent to
  the composition. (use a right triangle)
• cos(arcsin 2x)
• Tan(arccos x/3)

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Assignment

• Page 328
• 1-16, 17-31 odds, 39-66 every 3rd