Inverse Trigonometric

Functions 4.7

The inverse sine function, denoted by sin-1, is the inverse of the restricted sine function y = sin x, - /2 < x < / 2. Thus,y = sin-1 x means sin y = x,where - /2 < y < /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the inverse sine at x.”

y

-1

1

/2x

- /2

y = sin x

- /2 < x < /2

Domain: [- /2, /2]

Range: [-1, 1]

The Inverse Sine Function

Finding Exact Values of sin-1x

• Let = sin-1 x.• Rewrite step 1 as sin = x.• Use the exact values in the table to find the

value of in [-/2 , /2] that satisfies sin = x.

Example

6

2

1

6sin

2

1sin

2

1sin 1

• Find the exact value of sin-1(1/2)

Example

• Find the exact value of sin-1(-1/2)

6

21

6sin

21

sin

21

sin 1

The Inverse Cosine Function

The inverse cosine function,denoted by cos-1, is the inverse of the restricted cosine function

y = cos x, 0 < x < . Thus,y = cos-1 x means cos y = x,where 0 < y < and –1 < x < 1.

Find the exact value of cos-1 (-3 /2)

Text Example

65

23

65

cos

23

cos

23

cos 1

Find the exact value of cos-1 (2 /2)

Text Example

4

22

4cos

22

cos

22

cos 1

The Inverse Tangent Function

The inverse tangent function, denoted by tan-1, is the inverse of the restricted tangent function

y = tan x, -/2 < x < /2. Thus,y = tan-1 x means tan y = x,where - /2 < y < /2 and – < x < .

Find the exact value of tan-1 (-1)

Text Example

4

14

tan

1tan

1tan 1

Find the exact value of tan-1 (3)

Text Example

3

33

tan

3tan

3tan 1

Inverse Properties

The Sine Function and Its Inversesin (sin-1 x) = x for every x in the interval [-1, 1].sin-1(sin x) = x for every x in the interval [-/2,/2].

The Cosine Function and Its Inversecos (cos-1 x) = x for every x in the interval [-1, 1]. cos-1(cos x) = x for every x in the interval [0, ].

The Tangent Function and Its Inversetan (tan-1 x) = x for every real number x tan-1(tan x) = x for every x in the interval (-/2,/2).

Example

3.0coscos 1

23

sinsin 1 6.4coscos 1

Example

125

tansin 1

Example

31

sincot 1

Example

x1sincos

Using you Calculator

41

sin 1

Find the angle in radians to the nearest thousandth. Then find the angle in degree.

)7.43(tan 1

31

cos 1

25.10tan 1

Example• The following formula gives the viewing

angle θ, in radians, for a camera whose lens is x millimeters wide. Find the viewing angle in radians and degrees for a 28 millimeter lens.

28

634.21tan2 1

x

634.21tan2 1

4.753157.1 orradians