Question

Find the exact value of six trigonometrical functions of the angle $\theta$

Short answer

The simplifed value is$$\begin{gathered} \mathrm{si}n\theta =\frac{perpendicular}{hype\tan ous}=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2} \\ \cos \theta =\frac{base}{hype\tan ous}=\frac{2}{4}=\frac{1}{2} \\ \tan \theta =\frac{perpendicular}{base}=\frac{2\sqrt{3}}{2}=\sqrt{3} \\ cot\theta =\frac{1}{\tan \theta }=\frac{1}{\sqrt{3}} \\ csc\theta =\frac{1}{\sin \theta }=\frac{2}{\sqrt{3}} \\ sec\theta =\frac{1}{\cos \theta }=2 \end{gathered}$$

Step-by-step solution

Step 1. Given information    

The given data from the graph is$$\begin{gathered} Hypotenuse=4 \\ base=2 \end{gathered}$$

Step 2. Write all the trigonometrical function  

Here$$\begin{gathered} Hypotenuse=4 \\ base=2 \end{gathered}$$

, $$\begin{gathered} h^2=p^2+b^2 \\ {4}^2=p^2+2^2 \\ {4}^2-2^2=p^2 \\ 16-4=p^2 \\ {p}^2=12 \\ p=2\sqrt{3} \end{gathered}$$

role="math" localid="1646735032038" $$\begin{gathered} \mathrm{si}n\theta =\frac{perpendicular}{hype\tan ous}=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2} \\ \cos \theta =\frac{base}{hype\tan ous}=\frac{2}{4}=\frac{1}{2} \\ \tan \theta =\frac{perpendicular}{base}=\frac{2\sqrt{3}}{2}=\sqrt{3} \\ cot\theta =\frac{1}{\tan \theta }=\frac{1}{\sqrt{3}} \\ csc\theta =\frac{1}{\sin \theta }=\frac{2}{\sqrt{3}} \\ sec\theta =\frac{1}{\cos \theta }=2 \end{gathered}$$