1. ## sines

I need help proving these 2 identities.

1. If x = 18 degrees, prove that sin2x = cos3x. Find the exact values of sinx and cosx

2. If 2sin(x - y) = sin(x + y), prove that tanx = 3tan y

2. Originally Posted by Zetterbergx40
2. If 2sin(x - y) = sin(x + y), prove that tanx = 3tan y
$2sin(x - y) = sin(x + y)$

$2sin(x)cos(y) - 2sin(y)cos(x) = sin(x)cos(y) + sin(y)cos(x)$

$sin(x)cos(y) = 3sin(y)cos(x)$ <-- Divide both sides by $cos(x)cos(y)$

$tan(x) = 3tan(y)$

-Dan

3. Originally Posted by Zetterbergx40
1. If x = 18 degrees, prove that sin2x = cos3x. Find the exact values of sinx and cosx
I can't think of a way to help you on the first part, but for finding the exact value of sine and cosine:
$sin(2x) = cos(3x)$

$2sin(x)cos(x) = -4sin^2(x) cos(x) + cos(x)$ <-- Divide both sides by cos(x)

$2sin(x) = -4sin^2(x) + 1$

$4sin^2(x) + 2sin(x) - 1 = 0$

$sin(x) = \frac{-1 \pm \sqrt{5}}{4}$

Note that both solutions are acceptable, indicating that there are actually 4 solutions for x in $0 \leq x < 360$. Obviously the sin(18) will correspond to the positive solution.

From the sine equation we can find cosine:
$cos(x) = \pm \sqrt{1 - sin^2(x)}$
where the $\pm$ indicates which quadrant the angle is in. (So for 18 degrees, pick + because cosine is positive in the first quadrant.)

-Dan

4. ## Thanks Dan

Thanks alot, for the first reply, can you tell me what the solution would be in LS = RS form? Thanks

5. Hello, Zetterbergx40!

1. If $x = 18^o$, prove that: . $\sin2x \:= \:\cos3x$

We are asked to prove that: . $\sin36^o \:=\:\cos54^o$

The two angles are complementary and $\sin\theta \:=\:\cos(90^o - \theta)$

Or look at this triangle:
Code:
              *
/|
/ |
/  |
/36°|
c /    | b
/     |
/      |
/       |
/ 54°    |
*---------*
a