Use an addition or subtraction formula to find the solutions of the equation that are in the interval $[ 0 , \pi )$ . $\cos 8 t \cos 5 t = - \sin 8 t \sin 5 t$
A) $t = \frac { \pi } { 6 } , t = \frac { \pi } { 2 } , t = \frac { 5 \pi } { 6 }$
B) $t = \frac { 3 \pi } { 5 } , t = \frac { 4 \pi } { 5 }$
C) $t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$
D) $t = \frac { \pi } { 8 } , t = \frac { 3 \pi } { 8 } , t = \frac { 5 \pi } { 8 }$
E) $t = 0 , t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$

Question

Quizplus · Accepted Answer

Using the addition formula for cosine, we have$$\cos 8 t \cos 5 t = \frac{1}{2} (\cos 3 t + \cos 13 t)$$and$$-\sin 8 t \sin 5 t = \frac{1}{2} (\cos 13 t - \cos 3 t)$$So the equation becomes$$\frac{1}{2} (\cos 3 t + \cos 13 t) = \frac{1}{2} (\cos 13 t - \cos 3 t)$$which simplifies to$$\cos 3 t = 0$$Solving this equation in the interval $[ 0 , \pi )$, we get$$t = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}$$

Use an Addition or Subtraction Formula to Find the Solutions [0,π)[ 0 , \pi )[0,π)

Use an Addition or Subtraction Formula to Find the Solutions $[ 0 , \pi )$