Question

find the exact value of each of the six trigonometric functions of \\( \theta \\), if \\( (-8,-7) \\) is a point on the terminal side of angle \\( \theta \\).
\\\sin \theta=\frac{-7 \sqrt{113}}{113}\\(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\\\cos \theta=\frac{-8 \sqrt{113}}{113}\\(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\\\tan \theta=\frac{7}{8}\\(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)\\\csc \theta=\square\\(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall the definition of cosecant

The cosecant function is the reciprocal of the sine function, so \( \csc\theta=\frac{1}{\sin\theta} \).

Step2: Substitute the given value of \( \sin\theta \)

We know that \( \sin\theta = \frac{-7\sqrt{113}}{113} \), so we substitute this into the formula for \( \csc\theta \):
\( \csc\theta=\frac{1}{\frac{-7\sqrt{113}}{113}} \)

Step3: Simplify the reciprocal

To simplify \( \frac{1}{\frac{-7\sqrt{113}}{113}} \), we multiply the numerator and denominator by the reciprocal of the denominator, which is \( \frac{113}{-7\sqrt{113}} \). This gives us:
\( \csc\theta = 1\times\frac{113}{-7\sqrt{113}}=\frac{113}{-7\sqrt{113}} \)

Step4: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{113} \):
\( \csc\theta=\frac{113\sqrt{113}}{-7\times113} \)

Step5: Simplify the fraction

We can cancel out the common factor of 113 in the numerator and denominator:
\( \csc\theta = \frac{\sqrt{113}}{-7}=-\frac{\sqrt{113}}{7} \)

Answer:

\( -\frac{\sqrt{113}}{7} \)