Q. 16 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 16–23, find the exact value of each of the remaining trigonometric functions.

$\sin \theta =\frac{4}{5},\theta$ is acute.

Step-by-Step Solution

Answer

The exact value of $\sin \theta =\frac{4}{5}$ of the remaining trigonometric functions are,

$$\begin{gathered} 1. csc \theta =\frac{5}{4} \\ 2. \cos \theta =\frac{3}{5} \\ 3. sec \theta =\frac{5}{3} \\ 4. \tan \theta =\frac{4}{3} \\ 5. cot \theta =\frac{3}{4} \end{gathered}$$

Step 1 Given expression is,

$\sin \theta =\frac{4}{5}$.

Let $P=\left(x,y\right)$ is a point on the circle that corresponds to $\theta$. Since $\sin \theta =\frac{4}{5}>0$ and the angle $\theta$ is acute , therefore the point $P$ lies in the first quadrant.

Step 2 Let us consider y = 4 , r = 5 .

So, the point $P\left(x,4\right)$ corresponds to $\theta$ lies on the circle of radius $5$.

$$\begin{gathered} x^2+4^2=5^2 \\ {x}^2+16=25 \\ {x}^2=25-16 \\ {x}^2=\sqrt{9} \\ x=\pm 3 \end{gathered}$$

Step 3 Since P x , y lies in the first quadrant, we take y = 4 , x = 3 , r = 5 .

$$\begin{gathered} csc \theta =\frac{r}{y} \\ =\frac{5}{4} \\ \cos \theta =\frac{x}{r} \\ =\frac{3}{5} \\ sec \theta =\frac{r}{x} \\ =\frac{5}{3} \\ \tan \theta =\frac{y}{x} \\ =\frac{4}{3} \\ cot \theta =\frac{x}{y} \\ =\frac{3}{4} \end{gathered}$$

These are the remaining trigonometric functions of the given expression.