7.3: Inverse Trigonometric Functions | Precalculus

How it works

Inverse trigonometric functions return principal angles in restricted ranges. Arcsine returns angles from minus pi/2 to pi/2, arccosine from zero to pi, and arctangent from minus pi/2 to pi/2 excluding endpoints. They do not undo a trig function on every real angle.

\arcsin:[-1,1]\to[-\pi/2,\pi/2]

Worked examples

Example 1

Evaluate $\displaystyle \arcsin(1/2)$ .

The principal angle is $\displaystyle \pi/6$ .

Example 2

Evaluate $\displaystyle \arccos(-1/2)$ .

The value in $\displaystyle [0,\pi]$ is $\displaystyle 2\pi/3$ .

Problems

Problem 7.3.1

Evaluate $\displaystyle \arcsin(\sin(5\pi/6))$ .

Show worked solution

Sine is $\displaystyle 1/2$ , so the principal inverse is $\displaystyle \pi/6$ , not $\displaystyle 5\pi/6$ .

Problem 7.3.2

Evaluate $\displaystyle \arctan(-1)$ .

Show worked solution

The principal value is $\displaystyle -\pi/4$ .

Problem 7.3.3

Find $\displaystyle \cos(\arcsin(3/5))$ .

Show worked solution

The inverse sine angle is in quadrant I, so cosine is the positive value $\displaystyle 4/5$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 7.3.1

Problem 7.3.2

Problem 7.3.3

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