Question

Long 8. Prove the following geometrically (a) tan(180° - 0) = - tan 0​

Answer 1

Fortunately, I made some attachments when I was practicing trigonometry. Hope it helps. The measure of the angles is in radian.

Topic:

• Convention of Trigonometry

In mathematics, the convention of trigonometry made the calculation of trigonometric values other than acute angles. Trigonometric functions are defined on the unit circle.

Solution

First please take a view of the attachment named 'supplementary angles'.

① To prove it, compare the value of $\tan(\pi n-\theta)$ and $\tan(\theta)$. Since the axis of the symmetry is $y=0$,

$\implies \tan(\pi-\theta)=\dfrac{y}{-x}$

$\implies \tan(\theta)=\dfrac{y}{x}$

② Hence,

$\implies \tan(\pi-\theta)=-\tan(\theta)$

③ Therefore,

$\implies \tan(\pi-0)=-\tan(0)$

Some Exclusions

(Periodic) $\tan(\pi n+\theta)=\tan(\theta)$

The period of the function is $\pi$.