Inverse Trigonometric Functions Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
A right triangle has opposite side 55 and hypotenuse 1313. Find the angle θ\theta opposite the side of length 55 using arcsin⁡\arcsin.

Solution

  1. 1
    sin⁡θ=513\sin\theta = \frac{5}{13}.
  2. 2
    θ=arcsin⁡ ⁣(513)≈arcsin⁡(0.3846)≈22.6°\theta = \arcsin\!\left(\frac{5}{13}\right) \approx \arcsin(0.3846) \approx 22.6° (or ≈0.3948\approx0.3948 rad).

Answer

θ=arcsin⁡ ⁣(513)≈22.6°\theta = \arcsin\!\left(\dfrac{5}{13}\right) \approx 22.6°
Inverse sine converts a ratio (opposite/hypotenuse) back to an angle. This is the standard application of inverse trig in right triangle geometry, allowing us to find unknown angles from known side lengths.

About Inverse Trigonometric Functions

Functions that reverse the trigonometric functions: given a ratio, they return the corresponding angle. arcsin⁡\arcsin, arccos⁡\arccos, and arctan⁡\arctan are the inverses of sin⁡\sin, cos⁡\cos, and tan⁡\tan on restricted domains.

Learn more about Inverse Trigonometric Functions →

More Inverse Trigonometric Functions Examples

Example 1 easy

Evaluate [formula], [formula], and [formula]. State the range of each inverse trig function.

Example 2 hard

Simplify [formula] for [formula] without trigonometric functions in the final answer.

Example 4 medium

Solve [formula] on [formula] using [formula], and explain why there are two solutions.

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