Suppose that theta is an acute angle of a right triangle and that sec(0)= 5/2 Find cos(theta) and csc()

We are given that \(\sec(\theta) = \frac{5}{2}\), and we need to find \(\cos(\theta)\) and \(\csc(\theta)\).

### Step 1: Find \(\cos(\theta)\)

We know that the secant function is the reciprocal of the cosine function:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
So, we can express \(\cos(\theta)\) as:
\[
\cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{\frac{5}{2}} = \frac{2}{5}
\]

Thus, \(\cos(\theta) = \frac{2}{5}\).

### Step 2: Find \(\csc(\theta)\)

We know that the cosecant function is the reciprocal of the sine function:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
To find \(\sin(\theta)\), we use the Pythagorean identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Substitute \(\cos(\theta) = \frac{2}{5}\):
\[
\sin^2(\theta) + \left(\frac{2}{5}\right)^2 = 1
\]
\[
\sin^2(\theta) + \frac{4}{25} = 1
\]
\[
\sin^2(\theta) = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}
\]
\[
\sin(\theta) = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}
\]

Now, we can find \(\csc(\theta)\):
\[
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{\sqrt{21}} = \frac{5\sqrt{21}}{21}
\]

### Final Answers:
- \(\cos(\theta) = \frac{2}{5}\)
- \(\csc(\theta) = \frac{5\sqrt{21}}{21}\)

Do you want more details or have any questions?

Here are five related questions:
1. What is the value of \(\sin(\theta)\)?
2. How would you calculate \(\tan(\theta)\)?
3. Can you determine \(\cot(\theta)\) given \(\sec(\theta)\)?
4. What is the relationship between \(\sec(\theta)\) and \(\cos(\theta)\)?
5. How do we use the Pythagorean identity to find \(\sin(\theta)\)?

**Tip**: When working with trigonometric identities, it's always helpful to recall the basic reciprocal relationships between the functions (like \(\sec(\theta) = \frac{1}{\cos(\theta)}\)).

Suppose that theta is an acute angle of a right triangle and that sec(theta) = 5/2. Find cos(theta) and csc(theta).

Learn how to find cos(θ) and csc(θ) given that sec(θ) = 5/2 in a right triangle. This problem covers key trigonometric concepts such as reciprocal trigonometric functions and the Pythagorean identity. Suitable for students in Grades 9-12.

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