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
Please help!
I can't think of a way to help you on the first part, but for finding the exact value of sine and cosine:
$\displaystyle sin(2x) = cos(3x)$
$\displaystyle 2sin(x)cos(x) = -4sin^2(x) cos(x) + cos(x)$ <-- Divide both sides by cos(x)
$\displaystyle 2sin(x) = -4sin^2(x) + 1$
$\displaystyle 4sin^2(x) + 2sin(x) - 1 = 0$
Using the quadratic formula:
$\displaystyle sin(x) = \frac{-1 \pm \sqrt{5}}{4}$
Note that both solutions are acceptable, indicating that there are actually 4 solutions for x in $\displaystyle 0 \leq x < 360$. Obviously the sin(18) will correspond to the positive solution.
From the sine equation we can find cosine:
$\displaystyle cos(x) = \pm \sqrt{1 - sin^2(x)}$
where the $\displaystyle \pm$ indicates which quadrant the angle is in. (So for 18 degrees, pick + because cosine is positive in the first quadrant.)
-Dan
Hello, Zetterbergx40!
1. If $\displaystyle x = 18^o$, prove that: .$\displaystyle \sin2x \:= \:\cos3x$
We are asked to prove that: .$\displaystyle \sin36^o \:=\:\cos54^o$
The two angles are complementary and $\displaystyle \sin\theta \:=\:\cos(90^o - \theta)$
Or look at this triangle:Code:* /| / | / | /36°| c / | b / | / | / | / 54° | *---------* a