Question

Find the exact value of the six trigonometric functions of the angle $\theta$ in each figure.

Short answer

Required values of the trigonometric functions are:

$$\begin{gathered} \sin \theta =\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$c$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{5}$}\right. \\ \cos \theta =\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$c$}\right.=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{5}$}\right. \\ \tan \theta =\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ csc\theta =\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$b$}\right.=\raisebox{1ex}{$\sqrt{5}$}\!\left/ \!\raisebox{-1ex}{$1$}\right.=\sqrt{5} \\ sec\theta =\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$\sqrt{5}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ cot\theta =\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$1$}\right.=2 \end{gathered}$$

Step-by-step solution

Step 1. Given information

Let us consider the hypotenuse as 'c' , opposite as 'b' and the adjacent as 'a' .

Pythagorean Theorem:

$c^2=a^2+b^2$, where a, b and c are adjacent, opposite and hypotenuse respectively.

Trigonometric Functions:

$$\begin{gathered} \sin \theta =\frac{b}{c}csc\theta =\frac{c}{b} \\ \cos \theta =\frac{a}{c}sec\theta =\frac{c}{a} \\ \tan \theta =\frac{b}{a}cot\theta =\frac{a}{b} \end{gathered}$$

Step 2. Calculation

$$\begin{gathered} Given, \\ b=1,c=\sqrt{5} \\ ByPythagoreantheorem, \\ {c}^2=b^2+a^2 \\ {a}^2=c^2-b^2 \\ ={\left(\sqrt{5}\right)}^2-1^2 \\ =5-1 \\ =4 \\ a=\sqrt{4}=2 \\ now, \\ \sin \theta =\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$c$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{5}$}\right. \\ \cos \theta =\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$c$}\right.=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{5}$}\right. \\ \tan \theta =\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ csc\theta =\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$b$}\right.=\raisebox{1ex}{$\sqrt{5}$}\!\left/ \!\raisebox{-1ex}{$1$}\right.=\sqrt{5} \\ sec\theta =\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$a$}\right.=\raisebox{1ex}{$\sqrt{5}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ cot\theta =\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$1$}\right.=2 \end{gathered}$$