Question

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including allanswers in [0, 2) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is nosolution, indicate "No Solution."sin(3x) = V3cos(3x)

Answer 1

The given equation is:

`\sin (3x)=\sqrt[]{3}\cos (3x)`

Dividing both sides of the equation by cos(3x):

`(\sin(3x))/(\cos(3x))=\frac{\sqrt[]{3}\cos(3x)}{\cos(3x)}`

Cancel out the common term in the right hand side of the equation.

`\begin{gathered} (\sin(3x))/(\cos(3x))=\frac{\sqrt[]{3}\cancel{\textcolor{green}{\cos(3x)}}}{\cancel{\textcolor{green}{\cos(3x)}}} \\ (\sin(3x))/(\cos(3x))=\sqrt[]{3} \end{gathered}`
`\begin{gathered} \text{ Using the trigonometric identity }(\sin(3x))/(\cos(3x))=\tan (3x),\text{ it follows that} \\ \tan (3x)=\sqrt[]{3} \end{gathered}`

Therefore,

`3x=\tan ^(-1)(\sqrt[]{3})`

Hence,

`\begin{gathered} 3x=(\pi)/(3)+\pi n\: \\ \text{ Dividing both sides by }3\text{ we have} \\ x=(\pi)/(9)+(\pi n)/(3) \end{gathered}`

x = π / 9 + (πn)/3

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