Question

If $f\left(\theta \right)=\sin \theta$and $g\left(\theta \right)=\cos \theta$. Find the exact value of the function given below if $\theta =60°$. Do not use a calculator.

Function is$f\left(2\theta \right)$.

Short answer

The value of the function$f\left(2\theta \right)$is$\frac{\sqrt{3}}{2}$.

Step-by-step solution

Step 1. Given information.

Consider the given information$f\left(\theta \right)=\sin \theta$and the value of $60°$.

Step 2. Finding f2θ.

In trigonometric standard angel the value of $\sin 120°=\frac{\sqrt{3}}{2}$.

Substitute the value in function $f\left(2\theta \right)$.

$$\begin{gathered} f\left(2\theta \right)=f\left(2\times 60°\right) \\ =f\left(120°\right) \end{gathered}$$

Further simplify.

$$\begin{gathered} f\left(120°\right)=\sin 120° \\ =\sin \left(90°+30°\right) \\ =\cos 30° \\ =\frac{\sqrt{3}}{2} \end{gathered}$$