If f(\theta) = 3 tan\theta - sine2\theta, find f((\pi)/(6)). Do not use a calculator and express each exact value as a single fraction.

Question

If
`f(\theta) = 3 tan\theta - sine2\theta`, find
`f((\pi)/(6))`. Do not use a calculator and express each exact value as a single fraction.

Answer 1

`f(\pi/6)=√(3)-(√(3))/(2)`

Explanation:

Given
`f(\theta )=3tan(\theta)-sin(2\theta )`

As we know that
`tan(\theta )=(sin(\theta ))/(cos(\theta ))`

thus we can write

`f(\theta )=3* (sin(\theta ))/(cos(\theta ))-sin(2\theta )\\\\\therefore f(\pi /6 )=3* (sin(\pi /6))/(cos(\pi /6))-sin(2\cdot \pi/6)\\\\f(\pi /6)=3* (1/2)/(√(3)/2)-(√(3))/(2)`\

Thus

`f(\pi/6)=(3)/(√(3))-(√(3))/(2)\\\\f(\pi/6)=√(3)-(√(3))/(2)`