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 \( \tan C \), we start with the given values \( \cos C = \frac{5}{9} \) and \( \cot C = \frac{5}{\sqrt{56}} \).

### Step 1: Finding \( \sin C \) from \( \cos C \)

We use the Pythagorean identity:

\[
\sin^2 C + \cos^2 C = 1
\]

Substituting \( \cos C \):

\[
\sin^2 C + \left( \frac{5}{9} \right)^2 = 1
\]

Calculating \( \left( \frac{5}{9} \right)^2 \):

\[
\sin^2 C + \frac{25}{81} = 1
\]

Now, we isolate \( \sin^2 C \):

\[
\sin^2 C = 1 - \frac{25}{81} = \frac{81}{81} - \frac{25}{81} = \frac{56}{81}
\]

Thus, we take the positive root (since \( C \) is an acute angle):

\[
\sin C = \sqrt{\sin^2 C} = \sqrt{\frac{56}{81}} = \frac{\sqrt{56}}{9}
\]

### Step 2: Finding \( \sec C \)

Now we find \( \sec C \), which is the reciprocal of \( \cos C \):

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

### Step 3: Finding \( \tan C \)

We can find \( \tan C \) using the definitions of sine and cosine:

\[
\tan C = \frac{\sin C}{\cos C}
\]

Substituting the values we found:

\[
\tan C = \frac{\frac{\sqrt{56}}{9}}{\frac{5}{9}} = \frac{\sqrt{56}}{5}
\]

### Summary of Results

The values we have calculated are:

\[
\sec C = \frac{9}{5}
\]
\[
\tan C = \frac{\sqrt{56}}{5}
\]

Both values are boxed for clarity:

\[
\sec C = \frac{9}{5}, \quad \tan C = \frac{\sqrt{56}}{5}
\]