Question

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

Short answer

The simplified value is$$\begin{gathered} \mathrm{si}n\theta =\frac{perpendicular}{hype\tan ous}=\frac{4}{5} \\ \cos \theta =\frac{base}{hype\tan ous}=\frac{3}{5} \\ \tan \theta =\frac{perpendicular}{base}=\frac{4}{3} \\ cot\theta =\frac{1}{\tan \theta }=\frac{3}{4} \\ csc\theta =\frac{1}{\sin \theta }=\frac{5}{4} \\ sec\theta =\frac{1}{\cos \theta }=\frac{5}{3} \end{gathered}$$

Step-by-step solution

Step 1. Given information 

The given data from the graph is$$\begin{gathered} perpendicular=4 \\ base=3 \end{gathered}$$

Step 2. Write all the trigonometrical function 

Here, $$\begin{gathered} perpendicular=4 \\ base=3 \end{gathered}$$

By Pythagoras theorem,$$\begin{gathered} h^2=p^2+b^2 \\ =3^2+4^2 \\ =9+16 \\ =25 \\ h=5 \end{gathered}$$

role="math" localid="1646655843647" $$\begin{gathered} \mathrm{si}n\theta =\frac{perpendicular}{hype\tan ous}=\frac{4}{5} \\ \cos \theta =\frac{base}{hype\tan ous}=\frac{3}{5} \\ \tan \theta =\frac{perpendicular}{base}=\frac{4}{3} \\ cot\theta =\frac{1}{\tan \theta }=\frac{3}{4} \\ csc\theta =\frac{1}{\sin \theta }=\frac{5}{4} \\ sec\theta =\frac{1}{\cos \theta }=\frac{5}{3} \end{gathered}$$