Question

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

Answer 1

Answer:

$f(\pi/6)=\sqrt{3}-\frac{\sqrt{3}}{2}$

Step-by-step explanation:

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

As we know that $tan(\theta )=\frac{sin(\theta )}{cos(\theta )}$

thus we can write

$f(\theta )=3\times \frac{sin(\theta )}{cos(\theta )}-sin(2\theta )\\\\\therefore f(\pi /6 )=3\times \frac{sin(\pi /6)}{cos(\pi /6)}-sin(2\cdot \pi/6)\\\\f(\pi /6)=3\times \frac{1/2}{\sqrt{3}/2}-\frac{\sqrt{3}}{2}$\

Thus

$f(\pi/6)=\frac{3}{\sqrt{3}}-\frac{\sqrt{3}}{2}\\\\f(\pi/6)=\sqrt{3}-\frac{\sqrt{3}}{2}$