Q 102. - Precalculus Enhanced with Graphing Utilities

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 $g\left(2\theta \right)$.

Step-by-Step Solution

Answer

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

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 g 2 θ .

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.

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