Question

If$f\left(x\right)=\sin x,g\left(x\right)=\cos x$and$h\left(x\right)=2x,P\left(x\right)=\frac{x}{2}$.Find the value of$\left(f·h\right)\left(\frac{\pi }{6}\right)$.

Short answer

The value of$\left(f·g\right)\left(\frac{\pi }{6}\right)$is$\frac{\sqrt{3}}{4}$.

Step-by-step solution

Step 1. Given information.

Given $f\left(x\right)=\sin x,g\left(x\right)=\cos x$and $h=2x,P\left(x\right)=\frac{x}{2}$and the value of$x$is$\frac{\pi }{6}$.

Step 2. Simplify the given expression.

$\left(f·g\right)\left(\frac{\pi }{6}\right)$can be written as $f\left(\frac{\pi }{6}\right)$and $g\left(\frac{\pi }{6}\right)$.

Substitute $\sin \left(\frac{\pi }{6}\right)$for $f\left(\frac{\pi }{6}\right)$and $\cos \left(\frac{\pi }{6}\right)$for $g\left(\frac{\pi }{6}\right)$.

then,

$\left(f·g\right)\left(\frac{\pi }{6}\right)=\left(\sin \frac{\pi }{6}·\cos \frac{\pi }{6}\right)$

Step 3. Find the value of f·g π6.

In trigonometric standard angle the value of $\sin \left(\frac{\pi }{6}\right)$is $\frac{1}{2}$and the value of $\cos \left(\frac{\pi }{6}\right)$is $\frac{\sqrt{3}}{2}$.

Substitute the values in simplified expression.

$$\begin{gathered} \left(f·g\right)\left(\frac{\pi }{6}\right)=\left(\sin \frac{\pi }{6}·\cos \frac{\pi }{6}\right) \\ =\left(\frac{1}{2}·\frac{\sqrt{3}}{2}\right) \\ =\left(\frac{\sqrt{3}}{4}\right) \end{gathered}$$