Question

complete the table with exact trigonometric function values. do not use a calculator.

θ	sin θ	cos θ	tan θ	cot θ	sec θ	csc θ

θ	sin θ	cos θ	tan θ	cot θ	sec θ	csc θ

(simplify your answers, including any radicals. use integers or fractions for any numbers in the expressions.)

Explanation:

Step1: Calculate tanθ

Recall that \( \tan\theta=\frac{\sin\theta}{\cos\theta} \). Given \( \sin\theta = \frac{\sqrt{2}}{2} \) and \( \cos\theta=-\frac{\sqrt{2}}{2} \), substitute these values into the formula:
\( \tan\theta=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} \)
The \( \frac{\sqrt{2}}{2} \) terms cancel out, leaving \( \tan\theta=- 1 \).

Step2: Calculate cotθ

Recall that \( \cot\theta=\frac{\cos\theta}{\sin\theta} \) (or \( \cot\theta = \frac{1}{\tan\theta} \)). Using \( \cos\theta = -\frac{\sqrt{2}}{2} \) and \( \sin\theta=\frac{\sqrt{2}}{2} \):
\( \cot\theta=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} \)
The \( \frac{\sqrt{2}}{2} \) terms cancel out, leaving \( \cot\theta=-1 \). Also, using \( \cot\theta=\frac{1}{\tan\theta} \) with \( \tan\theta = - 1 \), we get \( \cot\theta=\frac{1}{-1}=-1 \).

Answer:

For \( \tan\theta \): \(-1\)
For \( \cot\theta \): \(-1\)