# How do you solve for x if cos (6x-20) = sin (2x - 10)?

Jul 24, 2018

$x = 15$

#### Explanation:

$\sin x = \cos \left(90 - x\right)$
$\cos x = \sin \left(90 - x\right)$

$\cos \left(6 x - 20\right) = \sin \left(90 - \left(6 x - 20\right)\right) = \sin \left(90 - 6 x + 20\right) = \sin \left(110 - 6 x\right)$

$\sin \left(110 - 6 x\right) = \sin \left(2 x - 10\right)$

$110 - 6 x = 2 x - 10$

$120 = 8 x$

$x = 15$

Sub $x = 15$ into the equation
LHS:
$\cos \left(6 x - 20\right)$
$= \cos \left(6 \times 15 - 20\right)$
$= \cos \left(90 - 20\right)$
$= \cos 70$

RHS:
$\sin \left(2 x - 10\right)$
$= \sin \left(2 \times 15 - 10\right)$
$= \sin \left(30 - 10\right)$
$= \sin 20$
$= \cos \left(90 - 20\right)$
$= \cos 70$
$= L H S$

Therefore, $x = 15$ is correct