Questions: QUESTION 16 Use the definition or identities to find the exact value of the indicated trigonometric function of the acute angle θ. sec θ = sqrt(10) Find cot θ. 1/3 3 sqrt(10)/10 sqrt(10)/3 sqrt(10)/10

Solution Steps

To find \(\cot \theta\) given \(\sec \theta = \sqrt{10}\), we can use trigonometric identities. First, recall that \(\sec \theta = \frac{1}{\cos \theta}\), so \(\cos \theta = \frac{1}{\sqrt{10}}\). Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can find \(\sin \theta\). Finally, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

We are given that \( \sec \theta = \sqrt{10} \).

Using the definition of secant, we have: \[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\sqrt{10}} \approx 0.3162 \]

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \( \sin \theta \): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{1}{\sqrt{10}}\right)^2 = 1 - \frac{1}{10} = \frac{9}{10} \] Thus, \[ \sin \theta = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \approx 0.9487 \]

Now, we can find \( \cot \theta \) using the definition: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{1}{\sqrt{10}}}{\frac{3}{\sqrt{10}}} = \frac{1}{3} \approx 0.3333 \]

Final Answer