Question

watch the video and then solve the problem given below. click here to watch the video. find the exact value of each of the six trigonometric functions of θ, if (-2, -3) is a point on the terminal side of angle θ.

Explanation:

Step1: Calculate the value of $r$

Given $x=-2$ and $y = - 3$, use the formula $r=\sqrt{x^{2}+y^{2}}$. So $r=\sqrt{(-2)^{2}+(-3)^{2}}=\sqrt{4 + 9}=\sqrt{13}$.

Step2: Calculate $\sin\theta$

By the definition $\sin\theta=\frac{y}{r}$, substituting $y=-3$ and $r = \sqrt{13}$, we get $\sin\theta=\frac{-3}{\sqrt{13}}=-\frac{3\sqrt{13}}{13}$.

Step3: Calculate $\cos\theta$

Using the definition $\cos\theta=\frac{x}{r}$, substituting $x=-2$ and $r=\sqrt{13}$, we have $\cos\theta=\frac{-2}{\sqrt{13}}=-\frac{2\sqrt{13}}{13}$.

Step4: Calculate $\tan\theta$

According to the definition $\tan\theta=\frac{y}{x}$, substituting $x=-2$ and $y=-3$, we obtain $\tan\theta=\frac{-3}{-2}=\frac{3}{2}$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{r}{y}$, substituting $r=\sqrt{13}$ and $y=-3$, we get $\csc\theta=-\frac{\sqrt{13}}{3}$.

Step6: Calculate $\sec\theta$

As $\sec\theta=\frac{r}{x}$, substituting $r=\sqrt{13}$ and $x=-2$, we have $\sec\theta=-\frac{\sqrt{13}}{2}$.

Step7: Calculate $\cot\theta$

Using the definition $\cot\theta=\frac{x}{y}$, substituting $x=-2$ and $y=-3$, we obtain $\cot\theta=\frac{2}{3}$.

Answer:

$\sin\theta=-\frac{3\sqrt{13}}{13}$, $\cos\theta=-\frac{2\sqrt{13}}{13}$, $\tan\theta=\frac{3}{2}$, $\csc\theta=-\frac{\sqrt{13}}{3}$, $\sec\theta=-\frac{\sqrt{13}}{2}$, $\cot\theta=\frac{2}{3}$