Use substitution to determine whether the given x-value is a solution of the equation.
Find All Solutions of a Trigonometric Equation
-$7 \cos x + 8 \sqrt { 2 } = 5 \cos x + 7 \sqrt { 2 }$
A) $x = \frac { 3 \pi } { 4 } + 2 n \pi$ or $x = \frac { 5 \pi } { 4 } + 2 n \pi$
B) $x = \frac { 3 \pi } { 4 } + n \pi$ or $x = \frac { 5 \pi } { 4 } + n \pi$
C) $x = \frac { \pi } { 4 } + 2 n \pi$ or $x = \frac { 7 \pi } { 4 } + 2 n \pi$
D) $x = \frac { \pi } { 4 } + n \pi$ or $x = \frac { 7 \pi } { 4 } + n \pi$

Question

Quizplus · Accepted Answer

First, simplify the given equation: $7 \cos x + 8 \sqrt { 2 } = 5 \cos x + 7 \sqrt { 2 }$. Subtract $5 \cos x$ and $7\sqrt{2}$ from both sides to get $2 \cos x = \sqrt{2}$. Divide both sides by 2 to get $\cos x = \frac{\sqrt{2}}{2}$. The solutions to $\cos x = \frac{\sqrt{2}}{2}$ are $x = \frac{3\pi}{4} + 2n\pi$ and $x = \frac{5\pi}{4} + 2n\pi$, where $n$ is any integer, because these are the angles in the unit circle where the cosine value is $\frac{\sqrt{2}}{2}$ and they repeat every $2\pi$ radians.

Use Substitution to Determine Whether the Given X-Value Is a Solution