Find the values of
`tan("cos"^(-1)(8)/(17))`

Answer

To find the value of \( \tan(\cos^{-1}(\frac{8}{17})) \), we will follow these steps:

### Step 1: Let \( \theta = \cos^{-1}(\frac{8}{17}) \)

This means that \( \cos(\theta) = \frac{8}{17} \).

### Step 2: Draw a right triangle

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Here, we can set:
- Adjacent side (base) = 8
- Hypotenuse = 17

### Step 3: Use the Pythagorean theorem to find the opposite side

According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ 17^2 = \text{Opposite}^2 + 8^2 \]
Calculating the squares:
\[ 289 = \text{Opposite}^2 + 64 \]
Now, subtract 64 from both sides:
\[ \text{Opposite}^2 = 289 - 64 = 225 \]
Taking the square root gives:
\[ \text{Opposite} = \sqrt{225} = 15 \]

### Step 4: Calculate \( \tan(\theta) \)

The tangent of an angle is defined as the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{15}{8} \]

### Step 5: Conclusion

Thus, we find that:
\[ \tan(\cos^{-1}(\frac{8}{17})) = \frac{15}{8} \]

### Final Answer:
\[ \frac{15}{8} \]

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