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$g\left(2\theta \right)$.

Short answer

The value of the function$g\left(2\theta \right)$is$\frac{-1}{2}$.

Step-by-step solution

Step 1. Given information .

Consider the given information $g\left(2\theta \right)$where $g\left(\theta \right)=\cos \theta$and the value of$\theta =60°$.

Step 2. Finding g2θ.

In the trigonometric standard angle the value of $\cos 120°=\frac{-1}{2}$.

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

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

Further simplify.

$$\begin{gathered} g\left(120°\right)=\cos 120° \\ =\cos \left(90°+30°\right) \\ =-\cos 30° \\ =-\frac{1}{2} \end{gathered}$$