# What is the inverse cosine of 1?

Aug 27, 2015

The inverse cosine of 1 is 0 (or any integer multiple of $2 \pi$)

#### Explanation:

Inverse cosine = arccos = cos^(-1)

If $\arccos \left(1\right) = \theta$
then $\cos \left(\theta\right) = 1 \textcolor{w h i t e}{\text{XXXX}}$by definition.

If $\cos \left(\theta\right) = 1$
then based on the unit circle definitions of trigonometric functions:
color(white)("XXX")"adjacent side"/"hypotenuse" = 1

i.e.color(white)("XX")"adjacent side" = "hypotenuse"

The only location where this is true is when $\theta = 0$ (or some multiple of full rotations that places the hypotenuse back on the positive X-axis).