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$\left(h·f\right)\left(\frac{5\pi }{6}\right)$.

Short answer

The value of$\left(h·f\right)\left(\frac{5\pi }{6}\right)$is$\frac{5\pi }{6}$.

Step-by-step solution

Step 1. Given information.

Given$h\left(x\right)=2x$role="math" localid="1646553814626" $,f\left(x\right)=\sin x$and the value of$x$is$\frac{5\pi }{6}$.

Step 2. Simplify the given expression.

$\left(h·f\right)\left(\frac{5\pi }{6}\right)$can be written as $h\left(\frac{5\pi }{6}\right)$and $f\left(\frac{5\pi }{6}\right)$.

Substitute $2x$for $h\left(\frac{5\pi }{6}\right)$and $\sin x$for $f\left(\frac{5\pi }{6}\right)$.

$\left(h·f\right)\left(\frac{5\pi }{6}\right)=\left(2\times \frac{5\pi }{6}·\sin \frac{5\pi }{6}\right)$

Step 3. Find the valuer of h·f 5π6.

In trigonometric standard angle the value of $\sin \frac{5\pi }{6}$is $\frac{1}{2}$.

Substitute the value in the simplified expression.

$$\begin{gathered} \left(h·f\right)\left(\frac{5\pi }{6}\right)=\left(2\times \frac{5\pi }{6}·\sin \frac{5\pi }{6}\right) \\ =\left(\frac{5\pi }{3}·\frac{1}{2}\right) \\ =\frac{5\pi }{6} \end{gathered}$$