Inverse Trigonometric Functions 4 7 The Inverse Sine

Inverse Trigonometric Functions 4. 7

The Inverse Sine Function 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 y = sin x - /2 < x < /2 1 - /2 -1 x Domain: [- /2, /2] Range: [-1, 1]

Finding Exact Values of -1 sin x • 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 • Find the exact value of sin-1(1/2)

Example • Find the exact value of sin-1(-1/2)

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.

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

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

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 < .

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

Text Example Find the exact value of tan-1 ( 3)

Inverse Properties The Sine Function and Its Inverse sin (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 Inverse cos (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 Inverse tan (tan-1 x) = x for every real number x tan-1(tan x) = x for every x in the interval (- /2, /2).

Example

Example

Example

Example

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

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.