Exact values using trig identities

**zweevu** · April 29th 2009, 11:30 AM

How would I find the exact value of sin(pie/16) using trig identities?

**TheEmptySet** · April 29th 2009, 12:04 PM

You will need the the identities

$\sin\left( \frac{\theta}{2}\right)=\pm \sqrt{\frac{1-\cos(\theta)}{2}}$

$\cos\left( \frac{\theta}{2}\right)=\pm \sqrt{\frac{1+\cos(\theta)}{2}}$

Note that

$\sin\left( \frac{\pi}{16}\right)=\sqrt{\frac{1-\cos\left( \frac{\pi}{8}\right)}{2}}$

and

$\cos\left( \frac{\pi}{8}\right)= \sqrt{\frac{1+\cos(\frac{\pi}{4})}{2}}$

I hope this helps

**Soroban** · April 29th 2009, 12:18 PM

Hello, zweevu!

We need two Double angle identities:

. . $\sin\theta \:=\:\sqrt{\frac{1-\cos2\theta}{2}}$

. . $\cos\theta \:=\:\sqrt{\frac{1+\cos2\theta}{2}}$

Find the exact value of $\sin\left(\frac{\pi}{16}\right)$ using trig identities.

$\sin\left(\frac{\pi}{16}\right) \;=\;\sqrt{\frac{1 - \cos(\frac{\pi}{8})}{2}}$

. . . $=\;\sqrt{\frac{1-\sqrt{\dfrac{1 + \cos(\frac{\pi}{4})}{2}}}{2}}$

. . . . $=\;\sqrt{\frac{1-\sqrt{\dfrac{1 + \frac{\sqrt{2}}{2} }{2}}}{2}}$

And I'll let you simplify it . . .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Well, okay . . . I got: . $\frac{\sqrt{2- \sqrt{2 + \sqrt{2}}}}{2}$

**zweevu** · April 29th 2009, 04:27 PM

Well that was simple. Thanks!

Thread: Exact values using trig identities

LinkBack

Thread Tools

Display

Exact values using trig identities

Similar Math Help Forum Discussions

trig exact values

exact value using trig identities

Trig Exact Values

Trig exact values

Trig identities, exact values

Search Tags