Trigonometry

**james_bond** · February 5th 2009, 01:41 AM

Show that $\sin\left[\frac{\pi}{14}\right]\cdot\sin\left[\frac{3\pi}{14}\right]\sin\left[\frac{5\pi}{14}\right]=\frac 18$ .

**AlvinCY** · February 5th 2009, 05:22 AM

Would this happen to be in the "complex numbers" section of the textbook/exercise? If so, did it have a first part? Also, is that the question? Or does it say "hence show that..."

Because I have seen these questions in our 4U textbooks (highest level maths in high school), and recognise them, I might be able to give you a hand if there is a first part!

**james_bond** · February 5th 2009, 06:48 AM

I'd appreciate your help. There are no other parts. This is the only question but I don't know from where did they get this.. It's between a collection of exercises xD and no complex numbers nearby.

**red_dog** · February 5th 2009, 11:52 AM

$8\sin\frac{\pi}{14}\sin\frac{3\pi}{14}\sin\frac{5\ pi}{14}=4\left(\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}\right)\sin\frac{5\pi}{14}=$

$2\left(2\sin\frac{5\pi}{14}\cos\frac{\pi}{7}-2\sin\frac{5\pi}{14}\cos\frac{2\pi}{7}\right)=2\le ft(\sin\frac{\pi}{2}+\sin\frac{3\pi}{14}-\sin\frac{9\pi}{14}-\sin\frac{\pi}{14}\right)=$

$=2\left(1+\sin\frac{3\pi}{14}-\sin\left(\pi-\frac{9\pi}{14}\right)-\sin\frac{\pi}{14}\right)=2\left(1+\sin\frac{3\pi} {14}-\sin\frac{5\pi}{14}-\sin\frac{\pi}{14}\right)$

Now I will show that $\sin\frac{3\pi}{14}-\sin\frac{5\pi}{14}-\sin\frac{\pi}{14}=-\frac{1}{2}$ (*)

This is equivalent to $\sin\frac{\pi}{14}-\sin\frac{3\pi}{14}+\sin\frac{5\pi}{14}=\frac{1}{2 }$

Multiply both members by $2\sin\frac{\pi}{7}$ then we transform products to sums.

$\cos\frac{\pi}{14}-\cos\frac{3\pi}{14}-\cos\frac{\pi}{14}+\cos\frac{5\pi}{14}+\cos\frac{3 \pi}{14}-\cos\frac{\pi}{2}=\sin\frac{\pi}{7}$

$\cos\frac{5\pi}{14}=\sin\frac{\pi}{7}$

$\cos\frac{5\pi}{14}=\cos\left(\frac{\pi}{2}-\frac{\pi}{7}\right)$

$\cos\frac{5\pi}{14}=\cos\frac{5\pi}{14}$

So, the identity (*) is true.

Thread: Trigonometry

LinkBack

Thread Tools

Display

Trigonometry

Similar Math Help Forum Discussions

Trigonometry to Memorize, and Trigonometry to Derive

How To Do Trigonometry For This...?

How To Do This Trigonometry

trigonometry

Trigonometry

Search Tags