Question

Angle C is an acute angle with cos C=5/9
and cot C=5/ root56
.

What are the values of sec C
and tan C
?

Answer 1

To find the values of \( \sec C \) and \( \tan C \), we can use the given values of \( \cos C \) and \( \cot C \).

1. **Finding \( \sec C \)**:
\[
\sec C = \frac{1}{\cos C}
\]
Given \( \cos C = \frac{5}{9} \), we have:
\[
\sec C = \frac{1}{\frac{5}{9}} = \frac{9}{5}
\]

2. **Finding \( \tan C \)**:
We know \( \cot C = \frac{1}{\tan C} \). Given \( \cot C = \frac{5}{\sqrt{56}} \), we can find \( \tan C \):
\[
\tan C = \frac{1}{\cot C} = \frac{\sqrt{56}}{5}
\]

Now we can summarize the results:
- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{\sqrt{56}}{5} \)

**Value Verification**:
To verify the values of \( \tan C \):
\[
\tan C = \frac{\sin C}{\cos C}
\]

First, we can find \( \sin C \) using the identity:
\[
\sin^2 C + \cos^2 C = 1
\]
Substituting \( \cos C \):
\[
\sin^2 C + \left( \frac{5}{9} \right)^2 = 1
\]
\[
\sin^2 C + \frac{25}{81} = 1
\]
\[
\sin^2 C = 1 - \frac{25}{81} = \frac{81}{81} - \frac{25}{81} = \frac{56}{81}
\]
\[
\sin C = \sqrt{\frac{56}{81}} = \frac{\sqrt{56}}{9}
\]

Now we calculate \( \tan C \):
\[
\tan C = \frac{\sin C}{\cos C} = \frac{\frac{\sqrt{56}}{9}}{\frac{5}{9}} = \frac{\sqrt{56}}{5}
\]

Both \( \sec C \) and \( \tan C \) have been verified.

**Final Results**:
- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{\sqrt{56}}{5} \)