Question

let (3, - 4) be a point on the terminal side of θ. find the exact values of cos θ, csc θ, and tan θ. cos θ = csc θ = tan θ =

Explanation:

Step1: Calculate the radius $r$

Given the point $(x = 3,y=-4)$ on the terminal - side of $\theta$, use the formula $r=\sqrt{x^{2}+y^{2}}$. So, $r=\sqrt{3^{2}+(-4)^{2}}=\sqrt{9 + 16}=\sqrt{25}=5$.

Step2: Calculate $\cos\theta$

The formula for $\cos\theta$ is $\cos\theta=\frac{x}{r}$. Substituting $x = 3$ and $r = 5$, we get $\cos\theta=\frac{3}{5}$.

Step3: Calculate $\csc\theta$

First, find $\sin\theta$. The formula for $\sin\theta$ is $\sin\theta=\frac{y}{r}$. Substituting $y=-4$ and $r = 5$, we have $\sin\theta=-\frac{4}{5}$. Since $\csc\theta=\frac{1}{\sin\theta}$, then $\csc\theta=-\frac{5}{4}$.

Step4: Calculate $\tan\theta$

The formula for $\tan\theta$ is $\tan\theta=\frac{y}{x}$. Substituting $x = 3$ and $y=-4$, we get $\tan\theta=-\frac{4}{3}$.

Answer:

$\cos\theta=\frac{3}{5}$
$\csc\theta=-\frac{5}{4}$
$\tan\theta=-\frac{4}{3}$