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Find the exact value of each of the remaining trigonometric functions of $\theta$. Rationalize denominators when applicable.
$\sin \theta=\frac{\sqrt{5}}{10}$, given that $\cos \theta< 0$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\cos \theta=$
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
B. The function is undefined.
Answer
\(\boxed{\cos \theta = -\frac{\sqrt{95}}{10}}\)
Step 1 ：Given that $\sin \theta = \frac{\sqrt{5}}{10}$, we can find $\sin^2 \theta$ as follows:
Step 2 ：$\sin^2 \theta = \left(\frac{\sqrt{5}}{10}\right)^2 = \frac{5}{100}$
Step 3 ：Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we can find $\cos^2 \theta$:
Step 4 ：$\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{5}{100} = \frac{95}{100}$
Step 5 ：Since $\cos \theta < 0$, we can find the value of $\cos \theta$ as follows:
Step 6 ：$\cos \theta = -\sqrt{\cos^2 \theta} = -\sqrt{\frac{95}{100}} = -\frac{\sqrt{95}}{10}$
Step 7 ：\(\boxed{\cos \theta = -\frac{\sqrt{95}}{10}}\)
