Solve the following equation for angles between 0 degrees and 360 degrees inclusive.
2 sin(square) x – 5 sin x cos x = 3 cos(square) x
do I solve it further like this:
2u (square) - 6uv +uv - 3v (square)
and then factorise?
After factorisation, I got:
sin x - 3 cos x = 0
or 2 sinx + cos x = 0
If I substitute, short equations like 7 cos x = 0 are coming up, through which it is not possible to deduce an answer mathematically.
Hello, Ilsa!
You're off to a good start . . .
$\displaystyle \text{Do I solve it further like this?}$
. . $\displaystyle 2u^2 - 5uv - 3v^2 \:=\:0\quad\Rightarrow\quad (u - 3v)(2u + v) \:=\:0$
$\displaystyle \text{Then: }\;\begin{Bmatrix}\sin x - 3\cos x &=& 0 & [1] \\
2\sin x + \cos x &=& 0 & [2] \end{Bmatrix}$
From [1] we have:
. . $\displaystyle \sin x \:=\:3\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x}\:=\:3 \quad\Rightarrow\quad \tan x \:=\:3$
Hence: .$\displaystyle x \:=\:\arctan 3 \quad\Rightarrow\quad \boxed{x\:\approx\: 71.6^o,\:251.6^o}$
From [2] we have:
. . $\displaystyle 2\sin x \:=\:-\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x} \:=\:\text{-}\frac{1}{2} \quad\Rightarrow\quad \tan x \:=\:\text{-}\frac{1}{2}$
Hence: .$\displaystyle x \:=\:\arctan(\text{-}0.5) \quad\Rightarrow\quad \boxed{x \:\approx\: 153.4^o,\:333.4^o}$