Question

Angle C is an acute angle with cos C=5/9
and cot C=5/sqrt(56)
.

What are the values of sec C
and tan C
?

Answer 1

To find the values of sec C and tan C given that \( \cos C = \frac{5}{9} \) and \( \cot C = \frac{5}{\sqrt{56}} \), we can use trigonometric identities.

### Step 1: Find \( \sec C \)

The secant function is the reciprocal of the cosine function:

\[
\sec C = \frac{1}{\cos C}
\]

Substituting the value of \( \cos C \):

\[
\sec C = \frac{1}{\frac{5}{9}} = \frac{9}{5}
\]

### Step 2: Find \( \tan C \)

The cotangent function gives us \( \tan C \) as the reciprocal of \( \cot C \):

\[
\tan C = \frac{1}{\cot C}
\]

Substituting the value of \( \cot C \):

\[
\tan C = \frac{1}{\frac{5}{\sqrt{56}}} = \frac{\sqrt{56}}{5}
\]

### Summary of Results

- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{\sqrt{56}}{5} \)

Finally, since \( \sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14} \), we can express \( \tan C \) more neatly:

\[
\tan C = \frac{2\sqrt{14}}{5}
\]

Thus, the final answers are:

- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{2\sqrt{14}}{5} \)