Question

1. -/6 points the point is on the terminal side of an angle in standard position. determine the exact values of the six trigonometric functions of the angle. (-5, -4) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = resources ebook

Explanation:

Step1: Calculate the radius $r$

Given the point $(x,y)=(-5,-4)$, use the formula $r = \sqrt{x^{2}+y^{2}}$. So $r=\sqrt{(-5)^{2}+(-4)^{2}}=\sqrt{25 + 16}=\sqrt{41}$.

Step2: Calculate $\sin\theta$

By the definition $\sin\theta=\frac{y}{r}$, substituting $y = - 4$ and $r=\sqrt{41}$, we get $\sin\theta=\frac{-4}{\sqrt{41}}=-\frac{4\sqrt{41}}{41}$.

Step3: Calculate $\cos\theta$

Using the definition $\cos\theta=\frac{x}{r}$, with $x=-5$ and $r = \sqrt{41}$, we have $\cos\theta=\frac{-5}{\sqrt{41}}=-\frac{5\sqrt{41}}{41}$.

Step4: Calculate $\tan\theta$

According to the definition $\tan\theta=\frac{y}{x}$, substituting $x=-5$ and $y = - 4$, we obtain $\tan\theta=\frac{-4}{-5}=\frac{4}{5}$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{r}{y}$, substituting $r=\sqrt{41}$ and $y=-4$, we get $\csc\theta=-\frac{\sqrt{41}}{4}$.

Step6: Calculate $\sec\theta$

Using the definition $\sec\theta=\frac{r}{x}$, with $r=\sqrt{41}$ and $x=-5$, we have $\sec\theta=-\frac{\sqrt{41}}{5}$.

Step7: Calculate $\cot\theta$

According to the definition $\cot\theta=\frac{x}{y}$, substituting $x=-5$ and $y=-4$, we obtain $\cot\theta=\frac{5}{4}$.

Answer:

$\sin\theta=-\frac{4\sqrt{41}}{41}$, $\cos\theta=-\frac{5\sqrt{41}}{41}$, $\tan\theta=\frac{4}{5}$, $\csc\theta=-\frac{\sqrt{41}}{4}$, $\sec\theta=-\frac{\sqrt{41}}{5}$, $\cot\theta=\frac{5}{4}$