Suppose that  θ  is an acute angle of a right triangle and that  sec(θ)=2.  Find  cot(θ)  and  cos(θ).

Given that \( \theta \) is an acute angle of a right triangle and \( \sec(\theta) = 2 \), we can find \( \cot(\theta) \) and \( \cos(\theta) \) using trigonometric identities.

### Step 1: Find \( \cos(\theta) \)
We know that the secant function is the reciprocal of the cosine function:

\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]

Given \( \sec(\theta) = 2 \), we can solve for \( \cos(\theta) \):

\[
\cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{2}
\]

So, \( \cos(\theta) = \frac{1}{2} \).

### Step 2: Find \( \cot(\theta) \)
To find \( \cot(\theta) \), we need to determine \( \sin(\theta) \), and then use the identity \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).

We can use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find \( \sin(\theta) \):

\[
\sin^2(\theta) + \left( \frac{1}{2} \right)^2 = 1
\]

\[
\sin^2(\theta) + \frac{1}{4} = 1
\]

\[
\sin^2(\theta) = 1 - \frac{1}{4} = \frac{3}{4}
\]

\[
\sin(\theta) = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}
\]

Now, using \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \):

\[
\cot(\theta) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

### Final Answer:
- \( \cos(\theta) = \frac{1}{2} \)
- \( \cot(\theta) = \frac{\sqrt{3}}{3} \)

Would you like further details or explanations? 

Here are 5 related questions for practice:
1. If \( \sin(\theta) = \frac{1}{2} \), what is \( \sec(\theta) \)?
2. Given \( \tan(\theta) = \frac{1}{2} \), find \( \cot(\theta) \).
3. If \( \cos(\theta) = \frac{\sqrt{2}}{2} \), what is \( \sin(\theta) \)?
4. How do you calculate \( \csc(\theta) \) if \( \sin(\theta) = \frac{3}{5} \)?
5. Given that \( \cot(\theta) = 2 \), find \( \tan(\theta) \).

**Tip:** Remember, secant and cosine are reciprocal trigonometric functions!

Suppose that θ is an acute angle of a right triangle and that sec(θ)=2. Find cot(θ) and cos(θ).

Solve for cot(θ) and cos(θ) when sec(θ)=2 in a right triangle. This problem explores key trigonometric identities, including the reciprocal functions sec(θ) and cot(θ), and uses the Pythagorean identity to solve for sine. Suitable for high school students in Grades 10-12.

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