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.

trigonometric identities 2

 

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.

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