1. ## Circular Functions Problem

How do I go about answering these two questions with relation to the attached image?
Find the values of the following:

a) $tan(\pi-\theta)$
b) $tan(-\theta)$

Thanks.

2. Hello user_5
Originally Posted by user_5
How do I go about answering these two questions with relation to the attached image?
Find the values of the following:

a) $tan(\pi-\theta)$
b) $tan(-\theta)$

Thanks.
Two things you need to know first:

• $\tan\theta = \frac{\sin\theta}{\cos\theta}$

• the co-ordinates of any point, P, on the unit circle are $(\cos\theta, \sin\theta)$ where $\theta$ is the angle through which OP has rotated anticlockwise from the positive $x$-axis.

So, in the diagram, $\cos\theta = \frac12$ and $\sin\theta=\frac{\sqrt3}{2}$ and so

$\tan\theta=\frac{\sqrt3}{2}\div\frac{1}{2}=\sqrt3$

OK, so how do we find $\tan(\pi-\theta)$?

• Look at the point on the unit circle where the angle is $(\pi-\theta)$.

• Work out its coordinates $(a,b)$ by reflecting the first point in the $y$-axis.

• Now write down $\cos(\pi-\theta)$ and $\sin(\pi - \theta)$

• Divide as I did above to give $\tan(\pi-\theta)$

Do $\tan(-\theta)$ in exactly the same way.

Can you complete it now?