Question

find the exact value of each of the six trigonometric functions of θ, if (5, - 5) is a point on the terminal side of angle θ.
sin θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
cos θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
tan θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
csc θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
sec θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
cot θ =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Calculate the radius $r$

Given the point $(x,y)=(5, - 5)$, use the formula $r=\sqrt{x^{2}+y^{2}}$. So $r = \sqrt{5^{2}+(-5)^{2}}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$.

Step2: Calculate $\sin\theta$

By the definition $\sin\theta=\frac{y}{r}$, substituting $y=-5$ and $r = 5\sqrt{2}$, we get $\sin\theta=\frac{-5}{5\sqrt{2}}=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}$.

Step3: Calculate $\cos\theta$

Using the definition $\cos\theta=\frac{x}{r}$, substituting $x = 5$ and $r=5\sqrt{2}$, we have $\cos\theta=\frac{5}{5\sqrt{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Step4: Calculate $\tan\theta$

By the definition $\tan\theta=\frac{y}{x}$, substituting $x = 5$ and $y=-5$, we obtain $\tan\theta=\frac{-5}{5}=-1$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{1}{\sin\theta}$, and $\sin\theta=-\frac{\sqrt{2}}{2}$, then $\csc\theta=-\sqrt{2}$.

Step6: Calculate $\sec\theta$

As $\sec\theta=\frac{1}{\cos\theta}$, and $\cos\theta=\frac{\sqrt{2}}{2}$, so $\sec\theta=\sqrt{2}$.

Step7: Calculate $\cot\theta$

Because $\cot\theta=\frac{1}{\tan\theta}$, and $\tan\theta=-1$, then $\cot\theta=-1$.

Answer:

$\sin\theta=-\frac{\sqrt{2}}{2}$
$\cos\theta=\frac{\sqrt{2}}{2}$
$\tan\theta=-1$
$\csc\theta=-\sqrt{2}$
$\sec\theta=\sqrt{2}$
$\cot\theta=-1$