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

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- Nov 16th 2010, 11:17 PMIlsaTrigonometric 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 - Nov 16th 2010, 11:30 PMjanvdl
- Nov 17th 2010, 01:41 AMIlsa
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. - Nov 17th 2010, 01:44 AMIlsa
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. - Nov 17th 2010, 02:58 AMSoroban
Hello, Ilsa!

You're off to a good start . . .

Quote:

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

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

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

- Nov 17th 2010, 08:12 PMIlsa
Thankyou, Mr. Soroban.

The solution you gave really helped.

I was substituting sinx = 3cosx into the next equation, which turned out be wrong.

Thankyou for your help!

(Happy)