13.4 Evaluate Inverse Trigonometric Functions

Inverse sine

If -1 £ a £ 1, then the inverse sine of a is an angle q, written q = sin^-1a, where sin q = a

and - £ q £ (or -90° £ q £ 90°).

Inverse cosine

If -1 £ a £ 1, then the inverse cosine of a is an angle q, written q = cos^-1a, where cos q = a and 0 £ 0 £ p (or 0° £ q £ 180°).

Inverse tangent

If a is any real number, then the inverse tangent of a is an angle q, written q = tan^-1a, where tan q = a and - £ q £ (or -90° £ q £ 90°).

INVERSE TRIGONOMETRIC FUNCTIONS

· If -1 £ a £ 1, then the _inverse sine_ of a is an angle q, written q = sin^-1 a, where sin q = a and - £ q £ (or -90° £ q £ 90°).

· If -1 £ a £ 1, then the _inverse cosine_ of a is an angle q, written q = cos^-1 a, where cos q = a and 0 £ q £ p (or 0° £ q £ 180°).

· If a is any real number, then the _inverse tangent_ of a is an angle q, written q = tan^-1 a, where tan and q - <q < (or -90° < q < 90°).

Example 1

Evaluate inverse trigonometric functions

Evaluate the expression in both radians and degrees.

a. cos^-13 b. tan^-1

Solution

a. There is _no angle_ whose cosine is 3. So, cos^-1 3 is _undefined _.

b. When - < q < or -90° < q < 90°, the angle whose tangent is is:

q = tan ^-1 = or q = tan ^-1 = _30°_

Example 2

Solve a trigonometric equation

Solve the equation cos q = where 270° < q < 360°.

Use a calculator to determine that in the interval 0° < q < 180°, the angle whose cosine is

is cos^-1 » __66.4°__. This angle is in Quadrant __1__. In Quadrant __IV__ (where 270° < q < 360°), the angle that has the same cosine value is:

q » _360°_ - _66.4_° = _293.6°_

Example 3

Find an angle measure

Find the measure of the angle q in the triangle shown.

Solution

In the right triangle, you are given the side opposite from q and the hypotenuse, so use the inverse sine function to solve for q.

sin q = = q = sin-1 » _22.6°_