Question

find the exact value of each of the six trigonometric functions of θ, if (9, −6) is a point on the terminal side of angle θ.
(sin \theta = \frac{-2sqrt{13}}{13})
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all de...
(cos \theta = \frac{3sqrt{13}}{13})
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all de...
(\tan \theta = \frac{-2}{3})
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all de...
(csc \theta = \frac{-sqrt{13}}{2})
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all d...
(sec \theta = square)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all d...

Explanation:

Step1: Recall the definition of secant

The secant function is the reciprocal of the cosine function, so $\sec\theta=\frac{1}{\cos\theta}$.

Step2: Substitute the value of $\cos\theta$

We know that $\cos\theta = \frac{3\sqrt{13}}{13}$. So we substitute this into the formula for $\sec\theta$:
$\sec\theta=\frac{1}{\frac{3\sqrt{13}}{13}}$

Step3: Simplify the reciprocal

To simplify $\frac{1}{\frac{3\sqrt{13}}{13}}$, we multiply the numerator and denominator by the reciprocal of the denominator, which is $\frac{13}{3\sqrt{13}}$. So we get $\frac{13}{3\sqrt{13}}$.

Step4: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{13}$:
$\frac{13\sqrt{13}}{3\times13}$

Step5: Simplify the fraction

The 13 in the numerator and denominator cancels out, leaving us with $\frac{\sqrt{13}}{3}$.

Answer:

$\frac{\sqrt{13}}{3}$