# Solve cos x - sin x tan x + 1 = 0 for 0°≤x≤360° ? need the solution asap, thank you.

May 22, 2018

$\text{ The Soln. Set.} \subset \left[{0}^{\circ} , {360}^{\circ}\right] = \left\{{60}^{\circ} , {180}^{\circ} , {300}^{\circ}\right\}$.

#### Explanation:

$\cos x - \sin x \tan x + 1 = 0$.

$\therefore \cos x - \sin x \cdot \sin \frac{x}{\cos} x + 1 + 0$.

$\therefore \frac{{\cos}^{2} x - {\sin}^{2} x + \cos x}{\cos} x = 0$.

$\therefore \left({\cos}^{2} x - {\sin}^{2} x\right) + \cos x = 0$.

$\therefore \cos 2 x + \cos x = 0$.

$\therefore 2 \cos \left(\frac{2 x + x}{2}\right) \cos \left(\frac{2 x - x}{2}\right) = 0$.

$\therefore \cos \left(3 \cdot \frac{x}{2}\right) \cos \left(\frac{x}{2}\right) = 0$.

Since, $\cos 3 \theta = 4 {\cos}^{3} \theta - 3 \cos \theta$, we have,

$\therefore \left\{4 {\cos}^{3} \left(\frac{x}{2}\right) - 3 \cos \left(\frac{x}{2}\right)\right\} \cos \left(\frac{x}{2}\right) = 0$.

$\therefore {\cos}^{2} \left(\frac{x}{2}\right) \left\{4 {\cos}^{2} \left(\frac{x}{2}\right) - 3\right\} = 0$.

$\therefore \cos \left(\frac{x}{2}\right) = 0 \mathmr{and} 4 {\cos}^{2} \left(\frac{x}{2}\right) = 3$.

$\therefore \cos \left(\frac{x}{2}\right) = 0 \mathmr{and} \cos \left(\frac{x}{2}\right) = \pm \frac{\sqrt{3}}{2}$.

Because, ${0}^{\circ} \le x \le {360}^{\circ} , {0}^{\circ} \le \frac{x}{2} \le {180}^{\circ}$.

$\therefore \cos \left(\frac{x}{2}\right) = 0 \Rightarrow \frac{x}{2} = {90}^{\circ} \Rightarrow x = {180}^{\circ}$.

Further, cos(x/2)=sqrt3/2 rArr x/2=30^@ rArr x=60^@, &,

$\cos \left(\frac{x}{2}\right) = - \frac{\sqrt{3}}{2} \Rightarrow \frac{x}{2} = {180}^{\circ} - {30}^{\circ} = {150}^{\circ} . \therefore x = {300}^{\circ}$.

$\therefore \text{ The Soln. Set.} \subset \left[{0}^{\circ} , {360}^{\circ}\right] = \left\{{60}^{\circ} , {180}^{\circ} , {300}^{\circ}\right\}$.

Feel the Joy of Maths.!

May 23, 2018

$\frac{\pi}{3} , \pi , \frac{5 \pi}{3}$

#### Explanation:

cos x - sin x.tan x = - 1
${\cos}^{2} x - {\sin}^{2} x = - \cos x$
$\cos 2 x = - \cos x = \cos \left(x + \pi\right)$
Unit circle and property of cos function give -->
$2 x = \pm \left(x + \pi\right)$
a. $2 x = x + \pi$
$x = \pi$
b. $2 x = - x - \pi$
$3 x = - \pi$, and $3 x = \pi$
1. $3 x = - \pi$ --> $x = - \frac{\pi}{3}$, or $x = \frac{5 \pi}{3}$ (co-terminal).
2. $3 x = \pi$ --> $x = \frac{\pi}{3}$
Check by calculator
$x = \frac{5 \pi}{3} = 300$ --> cos 300 = 0.5 --> sin 300 = -0.87 -->
tan 300 = - 1.732
cos x - sin x.tan x = 0.5 - (-0.87)(-1,732) = 0.5 - 1.5 = - 1. Proved