# explain why these two expressions are equal

• Aug 30th 2012, 03:25 PM
waleedrabbani
explain why these two expressions are equal
sin(9pi/7) = -sin(2pi/7)

Stuck....

Step by step would be appreciated very much.

Thanks
• Aug 30th 2012, 05:19 PM
skeeter
Re: explain why these two expressions are equal
note ...

$\displaystyle \sin\left(\frac{9\pi}{7}\right) = \sin\left(\pi + \frac{2\pi}{7}\right)$

... now use the sum identity for sine
• Aug 30th 2012, 05:37 PM
richard1234
Re: explain why these two expressions are equal
If you draw the terminal angles $\displaystyle \frac{9\pi}{7}$ and $\displaystyle \frac{2\pi}{7}$ on a unit circle, you will see that they differ by $\displaystyle \pi$. The sine of the angle is the y-coordinate of that angle/point on the unit circle. You will see that one is the negative of the other.

In general, $\displaystyle \sin (x + \pi) = -\sin x$.
• Aug 30th 2012, 07:38 PM
Prove It
Re: explain why these two expressions are equal
Quote:

Originally Posted by skeeter
note ...

$\displaystyle \sin\left(\frac{9\pi}{7}\right) = \sin\left(\pi + \frac{2\pi}{7}\right)$

... now use the sum identity for sine

Why use the sum identity? You already have it in the form \displaystyle \displaystyle \begin{align*} \sin{\left(\pi + \theta \right)} \equiv -\sin{(\theta)} \end{align*}.
• Aug 31st 2012, 03:11 AM
skeeter
Re: explain why these two expressions are equal
Quote:

Originally Posted by Prove It
Why use the sum identity? You already have it in the form \displaystyle \displaystyle \begin{align*} \sin{\left(\pi + \theta \right)} \equiv -\sin{(\theta)} \end{align*}.

so, you assume the OP is familiar with that specific identity derived from the sum identity?
• Aug 31st 2012, 03:12 AM
Prove It
Re: explain why these two expressions are equal
Quote:

Originally Posted by skeeter
so, you assume the OP is familiar with that specific identity derived from the sum identity?

While that identity can be derived from the sum identity, it is also blatantly obvious from the unit circle.
• Aug 31st 2012, 10:36 AM
skeeter
Re: explain why these two expressions are equal
Quote:

Originally Posted by Prove It
While that identity can be derived from the sum identity, it is also blatantly obvious from the unit circle.

"blatantly obvious from the unit circle" ... thanks for the lesson.
• Sep 7th 2012, 08:57 PM
trutinh
Re: explain why these two expressions are equal
i am appreciate you ... this is right .... work smart not hard...

\sin\left(\frac{9\pi}{7}\right) = \sin\left(\pi + \frac{2\pi}{7}\right)