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

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

Thanks.
Two things you need to know first:

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

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

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

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

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

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

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

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

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

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

Can you complete it now?