Question

Problem Solving

Find all of the solutions for the following equation.
tan theta= 4 tan theta - square 3

Answer 1

A)
`\theta=30^\circ+180n^\circ`

Explanation:

`tan\theta=4tan\theta-√(3)\\\\-3tan\theta=-√(3)\\\\9tan^2\theta=3\\\\tan^2\theta=(3)/(9)\\ \\tan^2\theta=(1)/(3)\\ \\(1-cos(2\theta))/(1+cos(2\theta))=(1)/(3)\\ \\ 3-3cos(2\theta)=1+cos(2\theta)\\\\2-3cos(2\theta)=cos(2\theta)\\\\2=4cos(2\theta)\\\\(2)/(4)=cos(2\theta)\\ \\(1)/(2)=cos(2\theta)\\ \\2\theta=60^\circ+360n^\circ\\\\\theta=30^\circ+180n^\circ`

Recall the identity
`tan^2(\theta)=(1-cos(2\theta))/(1+cos(2\theta))`