Question

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometricequation. Please identify all possible solutions by including all answers in [0, 21) and indicating theremaining answers by using n to represent any integer. Round your answer to four decimal places, ifnecessary. If there is no solution, indicate "No Solution."cos?(3x) = sin?(3x)

Answer 1

The given equation is

`\cos ^2(3x)=\sin ^2(3x),x\in\lbrack0,6\pi\rbrack`

Subtract sin^2(3x) from both sides

`\begin{gathered} \cos ^2(3x)-\sin ^2(3x)=\sin ^2(3x)-\sin ^2(3x) \\ \cos ^2(3x)-\sin ^2(3x)=0 \end{gathered}`

By using the identity

`\cos (2\theta)=\cos ^2(\theta)-\sin ^2(\theta)`

Then replace cits by 3x

`\begin{gathered} \cos ^2(3x)-\sin ^2(3x)=\cos (2*3x) \\ \cos ^2(3x)-\sin ^2(3x)=\cos (6x) \end{gathered}`

Then equate cos(6x) by 0

`\cos (6x)=0`

The value of cosine is equal to 0 at pi/2 and 3pi/2, then

`6x=(\pi)/(2)+\pi n`

Divide it by 6 to find x

`\begin{gathered} (6x)/(6)=((\pi)/(2))/(6)+(\pi n)/(6) \\ x=(\pi)/(12)+(\pi n)/(6) \end{gathered}`

n is an integer