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1 Inverse Trigonometry Functions and Their Derivatives

Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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Page 1: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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Inverse Trigonometry Functionsand Their Derivatives

Page 2: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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The graph of y = sin x does not pass the horizontal line test, so it has no inverse.

If we restrict the domain (to half a period), then we can talk about an inverse function.

Page 3: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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Definition

notation

EX 1 Evaluate these without a calculator.

a) c)

b) d)

Page 4: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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y = tan x y = sec x

Definition

[ ]

Page 5: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

5

EX 2 Evaluate without a calculator.

a) b) c)

Page 6: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

6

1

xθ

θ = cos-1x

sin(cos-1x) =

x

1θ sec (tan-1x) =

θ = tan-1x

x

1θ

θ = sec-1x

tan (sec-1x) =

Page 7: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

7

csc

EX 3 Calculate sin[2cos-1(1/4)] with no calculator.

Derivatives of Inverse Trig Functions

Let y = cos-1x. Find y'.

Page 8: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

8

C

C

EX 4

EX 5 Evaluate these integrals.

a)

b)

∈

Page 9: Inverse Trigonometry Functions and Their Derivatives · θ = sec-1x tan (sec-1x) = 7 csc EX 3 Calculate sin[2cos-1(1/4)] with no calculator. Derivatives of Inverse Trig Functions

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