Use Identities to Solve Trigonometric Equations
Solve the equation on the interval [0, 2π).
-$$\cos \left( x + \frac { \pi } { 3 } \right) + \cos \left( x - \frac { \pi } { 3 } \right) = 1$$
A) 0
B) $\frac { \pi } { 2 } , \frac { \pi } { 6 }$
C) $\frac { \pi } { 6 } , \frac { \pi } { 4 }$
D) $\frac { \pi } { 2 } , \frac { \pi } { 4 }$

Question

Quizplus · Accepted Answer

Using the sum-to-product formula, $\cos(A + B) + \cos(A - B) = 2 \cos A \cos B$, we can rewrite the given equation as $2\cos(x)\cos(\frac{\pi}{3}) = 1$. Since $\cos(\frac{\pi}{3}) = \frac{1}{2}$, the equation simplifies to $\cos(x) = \frac{1}{2}$. The solutions to $\cos(x) = \frac{1}{2}$ in the interval $[0, 2\pi)$ are $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$, but none of these solutions are listed in the options provided. The correct choice should reflect the solutions within the given interval, but since none of the options match, it seems there might have been a misunderstanding in interpreting the solutions or the options provided. Given the options, none directly match the correct solutions derived from the equation, suggesting a possible error in the question or answer choices.

Use Identities to Solve Trigonometric Equations Solve the Equation on the Interval

Use Identities to Solve Trigonometric Equations
Solve the Equation on the Interval