Inverse Trigonometric Functions Math Example 1

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

Example 1

easy
Evaluate arcsin⁡ ⁣(12)\arcsin\!\left(\frac{1}{2}\right), arccos⁡ ⁣(−22)\arccos\!\left(-\frac{\sqrt{2}}{2}\right), and arctan⁡(1)\arctan(1). State the range of each inverse trig function.

Solution

  1. 1
    arcsin⁡(12)\arcsin(\frac{1}{2}): find θ∈[−π/2,π/2]\theta\in[-\pi/2,\pi/2] with sin⁡θ=12\sin\theta=\frac{1}{2}. Answer: θ=π6\theta=\frac{\pi}{6}.
  2. 2
    arccos⁡(−22)\arccos(-\frac{\sqrt{2}}{2}): find θ∈[0,π]\theta\in[0,\pi] with cos⁡θ=−22\cos\theta=-\frac{\sqrt{2}}{2}. Answer: θ=3π4\theta=\frac{3\pi}{4} (Q2).
  3. 3
    arctan⁡(1)\arctan(1): find θ∈(−π/2,π/2)\theta\in(-\pi/2,\pi/2) with tan⁡θ=1\tan\theta=1. Answer: θ=π4\theta=\frac{\pi}{4}.

Answer

arcsin⁡(12)=π6\arcsin(\frac{1}{2})=\frac{\pi}{6}; arccos⁡(−22)=3π4\arccos(-\frac{\sqrt{2}}{2})=\frac{3\pi}{4}; arctan⁡(1)=π4\arctan(1)=\frac{\pi}{4}
Inverse trig functions return angles in their restricted ranges: arcsin⁡\arcsin in [−π/2,π/2][-\pi/2,\pi/2]; arccos⁡\arccos in [0,π][0,\pi]; arctan⁡\arctan in (−π/2,π/2)(-\pi/2,\pi/2). These restrictions ensure the inverse is a function (one-to-one).

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 2 hard

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

Example 3 easy

A right triangle has opposite side [formula] and hypotenuse [formula]. Find the angle [formula] oppo

Example 4 medium

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

← All Inverse Trigonometric Functions Examples Inverse Trigonometric Functions Concept