# Proof that sin(-θ) = – sin(θ), cos(-θ) = cos(θ), and tan(-θ) = -tan(θ)

In order to understand the proof below, you must have a prior knowledge about circular functions. The proof also uses symmetry which is I think quite basic for high schools students. The basic idea is that the point $(x,y)$ is symmetric with the point $(x,-y)$ with respect to the $x$ axis.

The angles $\theta$ and $-\theta$ are rotations with the positive x-axis as the initial side. The angle $\theta$ is a counterclockwise rotation and the angle $- \theta$ is a clockwise rotation as shown by the arrows in the diagram below.

Theorem

In this short post, we are going to prove the following trigonometric identities:

$\sin (-\theta) = - \sin \theta$

$\cos (- \theta) = \cos (\theta)$

$\tan (- \theta) = - \tan (\theta)$.

Proof

The terminal side of $\theta$ and $-\theta$ intersect the unit circle at points P and P’. The coordinates of

$P$ are $(\cos \theta, \sin \theta)$

$P'$ are $(\cos(- \theta), \sin (- \theta))$.

Since $P$ and $P'$ are symmetric with respect to the x-axis,

$\sin (\theta) = - \sin \theta$

$\cos (- \theta) = \cos \theta$.

Now, we know that

$\tan (- \theta) = \displaystyle \frac{\sin (- \theta)}{\cos(-\theta)}$

$= \displaystyle \frac{- \sin \theta}{\cos \theta}$

$= - \tan \theta$.

And we are done.