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.
Hello user_5Two 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?
Grandad