Question

find the exact values of all the trigonometric functions for the value of $\theta$.
the point $p$ is at $(3,-3\sqrt{3})$
$\sin\theta =$
$\cos\theta =$
$\tan\theta =$
$\cot\theta =$
$\sec\theta =$
$\csc\theta =$
question help: ebook message instructor

Explanation:

Step1: Calculate the radius \( r \)

For a point \( (x,y)=(3,-3\sqrt{3}) \) on the terminal side of an angle \( \theta \) in standard position, use the formula \( r=\sqrt{x^{2}+y^{2}} \).

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Step2: Calculate \( \sin\theta \)

Use the formula \( \sin\theta=\frac{y}{r} \).

$$ \sin\theta=\frac{-3\sqrt{3}}{6}=-\frac{\sqrt{3}}{2} $$

Step3: Calculate \( \cos\theta \)

Use the formula \( \cos\theta=\frac{x}{r} \).

$$ \cos\theta=\frac{3}{6}=\frac{1}{2} $$

Step4: Calculate \( \tan\theta \)

Use the formula \( \tan\theta=\frac{y}{x} \).

$$ \tan\theta=\frac{-3\sqrt{3}}{3}=-\sqrt{3} $$

Step5: Calculate \( \cot\theta \)

Use the formula \( \cot\theta=\frac{x}{y} \).

$$ \cot\theta=\frac{3}{-3\sqrt{3}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3} $$

Step6: Calculate \( \sec\theta \)

Use the formula \( \sec\theta=\frac{r}{x} \).

$$ \sec\theta=\frac{6}{3}=2 $$

Step7: Calculate \( \csc\theta \)

Use the formula \( \csc\theta=\frac{r}{y} \).

$$ \csc\theta=\frac{6}{-3\sqrt{3}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3} $$

Answer:

\(\sin\theta =-\frac{\sqrt{3}}{2}\), \(\cos\theta=\frac{1}{2}\), \(\tan\theta =-\sqrt{3}\), \(\cot\theta =-\frac{\sqrt{3}}{3}\), \(\sec\theta = 2\), \(\csc\theta=-\frac{2\sqrt{3}}{3}\)