Question

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

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

Short answer

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

role="math" localid="1646366252886" $$\begin{gathered} 1.csc\theta =\frac{5}{4} \\ 2.\cos \theta =\frac{3}{5} \\ 3.sec\theta =\frac{5}{3} \\ 4.\tan \theta =\frac{4}{3} \\ 5.cot\theta =\frac{3}{4} \end{gathered}$$

Step-by-step solution

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 Px,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.