Question

Find the values of six trigonometric functions for angle $\left(\theta \right)$ shown below.

Short answer

The six trigonometric functions are: $\sin \theta =\frac{1}{\sqrt{5}},\csc \theta =\sqrt{5},\cos \theta =\frac{2}{\sqrt{5}},\sec \theta =\frac{\sqrt{5}}{2},\tan \theta =\frac{1}{2},\cot \theta =2$.

Step-by-step solution

Step 1. Write down the given information.

The given figure is shown below.

Step 2. Concept used.

The six trigonometric functions are $\sin \theta , \csc \theta , \cos \theta , \sec \theta , \tan \theta , \cot \theta$.

Let $(x,y)$ be a point in X-Y plane and (r)be the length of a line segment drawn from the origin to the point.

In, $\Delta OAB,OA=r, OB=x, AB=y$

Therefore, six trigonometric functions are;

$\begin{array}{c}\sin \theta =\frac{AB}{OA}=\frac{y}{r},\csc \theta =\frac{1}{\sin \theta }=\frac{OA}{AB}=\frac{r}{y}\\ \cos \theta =\frac{OB}{OA}=\frac{x}{r},\sec \theta =\frac{1}{\cos \theta }=\frac{OA}{OB}=\frac{r}{x}\\ \tan \theta =\frac{AB}{OB}=\frac{y}{x},\cot \theta =\frac{1}{\tan \theta }=\frac{OB}{AB}=\frac{x}{y}\end{array}$

Step 3. Calculation.

From the given figure, the side adjacent to angle $\left(\theta \right)$is evaluated as:

$\begin{array}{c}\text{Adjacent}=\sqrt{{\left(2\sqrt{5}\right)}^2-2^2}\\ =4\end{array}$

Now, the six trigonometric functions are evaluated as:

localid="1648534524946" $\begin{array}{c}\sin \theta =\frac{2}{2\sqrt{5}}=\frac{1}{\sqrt{5}},\csc \theta =\sqrt{5}\\ \cos \theta =\frac{4}{2\sqrt{5}}=\frac{2}{\sqrt{5}},\sec \theta =\frac{\sqrt{5}}{2}\\ \tan \theta =\frac{1}{2},\cot \theta =2\end{array}$

Step 4. Conclusion.

The six trigonometric functions are: $\sin \theta =\frac{1}{\sqrt{5}},\csc \theta =\sqrt{5},\cos \theta =\frac{2}{\sqrt{5}},\sec \theta =\frac{\sqrt{5}}{2},\tan \theta =\frac{1}{2},\cot \theta =2$