# Thread: double angle/half angle formulas

1. ## double angle/half angle formulas

I have two problems for homework tonight that I can't get.
they are over double and half angles.
if someone can get any of these, and show how they did it or give a short explanation how, that would be awesome.

1. sin2x[4cos^4(x) - 4cos^2(x)]^2
I started by changing sin2x to 2sinxcosx & squared the parenthetical part
then I multiplied it all out but I don't see any way to simplify.

2. sin(C + D) = 7/8, (C+D) is in quadrant I, and sin C = 2/8, C is in quadrant I
find sin D
this one I'm rather lost. I know the formula for sin(C + D), so I can make it
(sinC)(cosD) + (cosC)(sinD) = 7/8
change the sin C to 2/8... but can't get it all figured out.

2. For (Q1),

$\sin(2x) \times (4 \cos^4x - 4 \cos^2x)^2$

$= 2 \sin{x} \cos{x} \times [4 \cos^2x ( \cos^2x - 1)]^2$

We know $1 - \cos^2x = \sin^2x$:

So $2 \sin{x} \cos{x} \times [4 \cos^2x ( \cos^2x - 1)]^2$

$= 2 \sin{x} \cos{x} \times (4 \cos^2x \times - \sin^2x)^2$

$= 2 \sin{x} \cos{x} \times (16 \cos^4x \, \sin^4x)$

$= 32 \sin^5x \, \cos^5x$

3. For (Q2),

$\sin(C + D) = \frac{7}{8}$

We know $\sin{C} = \frac{2}{8} = \frac{1}{4}$

$C = \arcsin{\left( \frac{1}{4} \right)} = 14.478_{}^{o}$ (Quadrant I)
$
\sin{(C + D)} = \sin{(14.478_{}^{o} + D)} = \frac{7}{8}$

$
{14.478_{}^{o} + D} = \arcsin{\left(\frac{7}{8} \right)} = 61.045_{}^{o}$
$D = 61.045_{}^{o} - 14.478_{}^{o} = 46.567_{}^{o}$
Now you can find $\sin{D}$