Inverse Trigonometric Ratios

A trigonometric function takes an angle and returns a ratio. Often the problem runs the other way: the ratio is known and the angle is what we are looking for. That reverse step is the job of the inverse trigonometric ratios — $\sin^{-1}$ , $\cos^{-1}$ and $\tan^{-1}$ .

Concept From a ratio back to an angle

Forward: angle ⟶ ratio, as in

\sin 30° = 0.5

Inverse: ratio ⟶ angle, as in

\sin^{-1}(0.5) = 30°

In the triangle above the sides give $\sin\theta = \dfrac{1}{2} = 0.5$ . The angle itself is not written anywhere, so we undo the sine:

\sin\theta = 0.5 \;\Longrightarrow\; \theta = \sin^{-1}(0.5) = 30°

Notation sin⁻¹ is not 1 ÷ sin

The −1 in $\sin^{-1}$ marks the inverse function, not an exponent. It does not mean $\dfrac{1}{\sin\theta}$ — that reciprocal is $\csc\theta$ , a completely different thing.

The same three ratios are also written $\arcsin$ , $\arccos$ and $\arctan$ . Both notations mean exactly the same: give me the angle whose ratio is this.

Example Finding the angle from the ratio

\sin\theta = \dfrac{1}{2}

\theta = \sin^{-1}\!\left(\dfrac{1}{2}\right)

⟹ θ = 30°

\cos\theta = \dfrac{\sqrt{2}}{2}

\theta = \cos^{-1}\!\left(\dfrac{\sqrt{2}}{2}\right)

⟹ θ = 45°

\tan\theta = \sqrt{3}

\theta = \tan^{-1}\!\left(\sqrt{3}\right)

⟹ θ = 60°

\cos\theta = 0.6

\theta = \cos^{-1}(0.6)

⟹ θ ≈ 53.13°

The first three come from the standard angles; the last one is not a standard angle, so the calculator gives a decimal.

Note The inverse returns the angle only

Knowing $\sin\theta = \dfrac{1}{2}$ fixes the angle at 30°, but it says nothing about how long the sides are. Every triangle below has the same ratio and therefore the same angle:

All three give $\theta = 30°$ . To pin down the actual side lengths you need one extra piece of information — at least one side.

Reference Input and output of each inverse

Ratio	Symbol	Input	Angle returned
Inverse sine	sin⁻¹	−1 to 1	−90° to 90°
Inverse cosine	cos⁻¹	−1 to 1	0° to 180°
Inverse tangent	tan⁻¹	any real number	−90° to 90°

Note On the calculator

Set the mode to Degrees, not radians — otherwise $\sin^{-1}(0.5)$ reads 0.5236 instead of 30°.

Feeding $\sin^{-1}$ or $\cos^{-1}$ a number outside −1 to 1 is an error, because no angle has such a ratio. $\tan^{-1}$ has no such limit.

Summary

An inverse trigonometric ratio takes the ratio and returns the angle — the exact reverse of the original function.
The −1 marks the inverse function, not a power: sin⁻¹ is not 1 ÷ sin, which is csc.
The result is an angle only; finding the side lengths needs at least one given side.
sin⁻¹ and cos⁻¹ accept inputs from −1 to 1 only; tan⁻¹ accepts any real number. Keep the calculator in Degrees mode.