Question

How do I find the following trigonometric ratios:1. Cos theta2.Cos alpha For this image below

Answer 1

from the diagram above, we can find the various trigonometric ratios using a system call SOHCATOA

$\begin{gathered} \text{SOHCAHTOA} \\ \text{ sin}\theta=\frac{opposite}{hypothenus} \\ \text{cos}\theta=\frac{adjacent}{hypothenus} \\ \text{tan}\theta=\frac{opposite}{adjacent} \end{gathered}$$\begin{gathered} \text{hypothenus}=4.36 \\ \text{opposite}=2.46 \\ \text{adjacent}=3.6 \end{gathered}$

so, we can go ahead and plug in the variables into each expression

$\begin{gathered} \text{ sin}\theta=\frac{opp}{hyp} \\ \text{ sin}\theta=\frac{2.46}{4.36} \\ \text{ sin }\theta=0.5642 \\ \text{take the sine inverse } \\ \theta=\sin ^{-1}0.5642 \\ \theta=34.35^0 \end{gathered}$$\begin{gathered} \text{cos}\theta=\frac{adj}{hyp} \\ \text{cos}\theta=\frac{3.6}{4.36} \\ cos\theta=0.8257 \\ \theta=\cos ^{-1}0.8257 \\ \theta=34.35^0 \end{gathered}$$\begin{gathered} \text{cos}\alpha=\cos (90-\theta) \\ \text{cos}\alpha=\cos (90-34.35) \\ \text{cos}\alpha=\cos 55.65 \\ \text{cos}\alpha=55.65^0 \end{gathered}$