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 $\sec(\theta) = \frac{1}{\cos(\theta)}$).