NotesByLex

Inverse Trigonomentric Functions

Lex Toumbourou ·

For each of the 3 trigonometric functions, there is a function that can invert the results.

For example, the Sine, can be expressed as follows

sin⁡(θ)=x\sin(\theta) = x

Is there a function that can take xx and return θ\theta? It's called arcsin⁡\arcsin

arcsin⁡(x)=θ\arcsin(x) = \theta

We have the following functions:

  • arcsin⁡(x)\arcsin(x) or sin⁡−1(x)\sin^{-1}(x) is the inverse of sin⁡(x)\sin(x)
  • arccos⁡(x)\arccos(x) or cos⁡−1(x)\cos^{-1}(x) is the inverse of cos⁡(x)\cos(x)
  • arctan⁡(x)\arctan(x) or tan⁡−1(x)\tan^{-1}(x) is the inverse of tan⁡(x)\tan(x)

Trig functions aren't really invertible (see Inverse Function) as they have multiple inputs with the same output (see Injective Function), like sin⁡(0)=sin⁡(π)=0\sin(0) = \sin(\pi) = 0.

So how can we define sin⁡−1(0)\sin^{-1}(0)?

To do that, we have to restrict the domain of the original functions to an interval where they are invertible. These domains determine the range of inverse functions.

The value from the appropriate functions is called the Principal value of the function:

Function Radians Degrees
arcsin⁡\arcsin −π2≤arcsin⁡(θ)≤π2-\frac{\pi}{2} \le \arcsin(\theta) \le \frac{\pi}{2} 90°≤arcsin⁡(θ)≤90°90° \le \arcsin(\theta) \le 90°
arccos⁡\arccos 0≤arccos⁡(θ)≤π0 \le \arccos(\theta) \le \pi 0°≤arccos⁡(θ)≤180°0° \le \arccos(\theta) \le 180°
arctan⁡\arctan −π2<arctan⁡(θ)<pi2-\frac{\pi}{2} \lt \arctan(\theta) \lt \frac{pi}{2} −90°<arctan⁡(θ)<90°-90° \lt \arctan(\theta) \lt 90°

If the theta is outside of