Q. 84 - Precalculus Enhanced with Graphing Utilities

Question

A point on the terminal side of an angle $\theta$ in standard position is given. Find the exact value of each of the six trigonometric functions of $\theta$ : $(0.3, 0.4)$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions are, 

$$\begin{gathered} \sin \theta = \frac{0.4}{0.5} \\ \cos \theta =\frac{0.3}{0.5} \\ \tan \theta =\frac{0.4}{0.3} \\ \csc \theta =\frac{0.5}{0.4} \\ \sec \theta =\frac{0.5}{0.3} \\ \cot \theta =\frac{0.3}{0.4} \end{gathered}$$

Step 1. Given information

We are given a point $(0.3, 0.4)$

We need to find the exact value of each of the six trigonometric functions.

Step 2. Concept

Let $(x,y)$ be a point on the terminal side of the angle $\theta$ in the standard position, which is also on the circle $x^2+y^2=r^2$. Then,

$$\begin{gathered} \sin \theta = \frac{y}{r} \\ \cos \theta =\frac{x}{r} \\ \tan \theta =\frac{y}{x} \\ \csc \theta = \frac{r}{y} \\ \sec \theta = \frac{r}{x} \\ \cot \theta = \frac{x}{y} \end{gathered}$$ 

Step 3. Finding the value of r

The given point is $(0.3, 0.4)$. Here, $x=0.3, y=0.4$

To find $r$, we have

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{(0.3)}^2+{(0.4)}^2} \\ r=\sqrt{0.09+0.16} \\ r=\sqrt{0.25} \\ r=0.5 \end{gathered}$$

Step 4. To find the values of six trigonometric functions

The given point is $(0.3, 0.4)$. Here, $x=0.3, y=0.4$

We know that $r=0.5$

Then the values of the six trigonometric functions of $\theta$ are,

$$\begin{gathered} \sin \theta = \frac{y}{r}=\frac{0.4}{0.5} \\ \cos \theta =\frac{x}{r}=\frac{0.3}{0.5} \\ \tan \theta =\frac{y}{x}=\frac{0.4}{0.3} \\ \csc \theta = \frac{r}{y}=\frac{0.5}{0.4} \\ \sec \theta = \frac{r}{x}=\frac{0.5}{0.3} \\ \cot \theta = \frac{x}{y}=\frac{0.3}{0.4} \end{gathered}$$

Step 5. Final Answer

The values of the six trigonometric functions of $\theta$ are

$$\begin{gathered} \sin \theta = \frac{0.4}{0.5} \\ \cos \theta =\frac{0.3}{0.5} \\ \tan \theta =\frac{0.4}{0.3} \\ \csc \theta =\frac{0.5}{0.4} \\ \sec \theta =\frac{0.5}{0.3} \\ \cot \theta =\frac{0.3}{0.4} \end{gathered}$$