Question

PRE-CALCULUS H 1: MA350 601 PRE-CALCULUS H 1
05.02 Solving Trigonometric Equations

Question 3
Find all solutions to the equation.
cos2x + 2 cos x + 1 = 0

Answer 1

Given to solve,

$\displaystyle\sf{\longrightarrow \cos(2x)+2\cos x+1=0}$

$\displaystyle\sf{\longrightarrow 2\cos^2x-1+2\cos x+1=0}$

$\displaystyle\sf{\longrightarrow 2\cos^2x+2\cos x=0}$

$\displaystyle\sf{\longrightarrow 2\cos x(\cos x+1)=0}$

$\displaystyle\sf{\Longrightarrow \cos x=0\quad OR\quad \cos x=-1}$

$\displaystyle\sf{\Longrightarrow\underline {\underline {x=\dfrac {(2n+1)\pi}{2}\quad OR\quad x=(2n+1)\pi}}}$

$\displaystyle\sf {\Longrightarrow\underline {\underline {x\in\left\{\dfrac {n\pi}{2}\right\}-\left\{2n\pi\right\}}}}$

where $\displaystyle\sf {n\in\mathbb{Z}.}$