Q.90 - Precalculus Enhanced with Graphing Utilities

Question

if $\cos \left(\theta \right)=0.2$, then find the value of $\cos \left(\theta \right)+\cos \left(\theta +2\pi \right)+\cos \left(\theta +4\pi \right)$

Step-by-Step Solution

Answer

The exact value of $\cos \left(\theta \right)+\cos \left(\theta +2\pi \right)+\cos \left(\theta +4\pi \right) is 0.6$

Step 1: Given information

We are given that $\cos \left(\theta \right)=0.2$

Step 2: Use periodicity of trigonometric function cos θ + 2 π = cos θ

We get,

$$\begin{gathered} \cos \left(\theta \right)+\cos \left(\theta +2\pi \right)+\cos \left(\theta +4\pi \right) \\ \cos \left(\theta \right)+\cos \left(\theta \right)+\cos \left(\theta +2\pi \right) \\ \cos \left(\theta \right)+\cos \left(\theta \right)+\cos \left(\theta \right) \\ 3\cos \left(\theta \right) \\ =3(0.2) \\ =0.6 \end{gathered}$$

Step 3: Conclusion

The value of $\cos \left(\theta \right)+\cos \left(\theta +2\pi \right)+\cos \left(\theta +4\pi \right) is 0.6$