# Trig Equations with Multiple Trig Functions help

• Apr 6th 2008, 05:37 PM
~berserk
Trig Equations with Multiple Trig Functions help
Find, to the nearest tenth of a degree, all values of x in the interval 0≤x≤360 that satisfy the equation.

tan x=cos x

also solve for x in the interval 0≤x≤2π

cos ½x= cos x
• Apr 6th 2008, 05:48 PM
o_O
$\displaystyle \tan x = \cos x$
$\displaystyle \tan x - \cos x = 0$
$\displaystyle \frac{\sin x}{\cos x} - \cos x = 0$
$\displaystyle \sin x - \cos^{2} x = 0 \quad \quad \mbox{Multiply both sides by } \cos x$
$\displaystyle \sin x - \left(1 - \sin^{2} x\right) = 0$
$\displaystyle \sin^{2} x + \sin x - 1 = 0$

Solve the quadratic for sin x and solve for x.

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$\displaystyle \cos \left(\frac{1}{2}x\right) = \cos x$
$\displaystyle \cos^{2}\left(\frac{1}{2}x\right) = \cos^{2} x$
$\displaystyle \frac{1}{2} \left(1 + \cos x\right) = \cos^{2} x \quad \quad \mbox{Half-angle Identity}$
$\displaystyle 0 = \cos^{2} x - \frac{1}{2} \cos x - \frac{1}{2}$
$\displaystyle 0 = 2 \cos^{2} x - \cos x - 1$

You know the dril.