exact value question

**Hydralore** · December 18th 2007, 11:34 AM

If cos (-) = 4/5 , 0 < (-) < pie/2

find the exact value of

(a) sin 2(-)
(b) sin 4(-)

given that tan x = square root of 11 all over 3, o < x < pie/2 , find the exact value of sin x

If anyone could help me with these questions id be gratefull

edit : damn.. can anyone help me show that sin (2x + x) = 3sin x - 4sin(to the power of 3)x

**bobak** · December 18th 2007, 11:41 AM

$\cos \theta = \frac{4}{5}$

using $\sin^2 \theta + \cos^2 \theta = 1$ or and appropriate triangle calculate the exact value of $\sin \theta$ .

you should now that $\sin 2 \theta = 2\sin \theta \cos \theta$
this should be an easy calculation.

similarly $\sin 4 \theta = 2\sin 2 \theta \cos 2 \theta$
should be able to take it form there

**Hydralore** · December 18th 2007, 11:44 AM

Thanks,

i also thought i could do another question but i need a little help.

show that sin (2x + x) = 3 sin x - 4 sin^3x

**bobak** · December 18th 2007, 11:54 AM

show that $\sin (2\theta + \theta) = 3 \sin \theta - 4 \sin^3 \theta$

starting on the LHS using the compound angle formula
$\sin (2\theta + \theta) = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta$

you know that $\sin 2\theta = 2\sin \theta \cos \theta$
and you should know $\cos 2\theta = cos^2 \theta - \sin^2 \theta$

you may need to use $\cos^2 \theta = 1 - \sin^2 \theta$

**Krizalid** · December 18th 2007, 03:00 PM

Originally Posted by Hydralore

can anyone help me show that sin (2x + x) = 3sin x - 4sin(to the power of 3)x

$\sin 3x = \text{Im} \left( {e^{ix} } \right)^3 = \text{Im} \left( {\cos x + i\sin x} \right)^3 .$

Now

$\left( {\cos x + i\sin x} \right)^3 = \cos ^3 x + 3i\sin x\cos ^2 x - 3\sin ^2 x\cos x - i\sin ^3 x.$

And finally

$\sin 3x = 3\sin x(1 - \sin ^2 x) - \sin ^3 x = 3\sin x - 4\sin ^3 x\,\blacksquare$