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$\theta$:$(0.3,0.4)$

Short 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-by-step solution

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,

role="math" localid="1646760380391" $$\begin{gathered} \sin \theta =\frac{y}{r}=\frac{0.4}{0.5} \\ \cos \theta =\frac{x}{r}=\frac{0.3}{0.5} \\ \tan \theta =\frac{y}{x}=\frac{0.4}{0.3} \\ \csc \theta =\frac{r}{y}=\frac{0.5}{0.4} \\ \sec \theta =\frac{r}{x}=\frac{0.5}{0.3} \\ \cot \theta =\frac{x}{y}=\frac{0.3}{0.4} \end{gathered}$$

Step 5. Final Answer

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

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