1. ## Trigonometric equations

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

2. Originally Posted by Ilsa
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
$2 sin^2 x - 5 sin x cos x - 3 cos^2 x = 0$

What if we let $sinx = u$ and $cos x = v$ ?

$2u^2 - 5uv - 3v^2 = 0$

Can you continue?

3. 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.

4. the answers are supposed to be: 71.6 degrees, 153.4 degrees, 251.6 degrees and 333.4 degrees.
However, I am not able to solve the equation and attain those answers.

5. Hello, Ilsa!

You're off to a good start . . .

$\text{Do I solve it further like this?}$

. . $2u^2 - 5uv - 3v^2 \:=\:0\quad\Rightarrow\quad (u - 3v)(2u + v) \:=\:0$

$\text{Then: }\;\begin{Bmatrix}\sin x - 3\cos x &=& 0 & [1] \\
2\sin x + \cos x &=& 0 & [2] \end{Bmatrix}$

From [1] we have:
. . $\sin x \:=\:3\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x}\:=\:3 \quad\Rightarrow\quad \tan x \:=\:3$

Hence: . $x \:=\:\arctan 3 \quad\Rightarrow\quad \boxed{x\:\approx\: 71.6^o,\:251.6^o}$

From [2] we have:
. . $2\sin x \:=\:-\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x} \:=\:\text{-}\frac{1}{2} \quad\Rightarrow\quad \tan x \:=\:\text{-}\frac{1}{2}$

Hence: . $x \:=\:\arctan(\text{-}0.5) \quad\Rightarrow\quad \boxed{x \:\approx\: 153.4^o,\:333.4^o}$

6. Thankyou, Mr. Soroban.
The solution you gave really helped.
I was substituting sinx = 3cosx into the next equation, which turned out be wrong.