If `sin theta = (8)/(17)` then the value of cos `theta and tan theta` is

Options

A. `(15)/(17),(8)/(15)`
B. `(15)/(8),(8)/(15)`
C. `(15)/(17),(15)/(8)`
D. None of these

Answer

To solve the problem where \( \sin \theta = \frac{8}{17} \) and we need to find the values of \( \cos \theta \) and \( \tan \theta \), we can follow these steps:

### Step 1: Understand the relationship of sine in a right triangle
Given that \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), we can identify the opposite side (perpendicular) and the hypotenuse from the given sine value. Here, the opposite side is 8 and the hypotenuse is 17.

### Step 2: Use the Pythagorean theorem to find the adjacent side (base)
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Substituting the known values:
\[ 17^2 = 8^2 + \text{adjacent}^2 \]
Calculating the squares:
\[ 289 = 64 + \text{adjacent}^2 \]
Now, isolate the adjacent side:
\[ \text{adjacent}^2 = 289 - 64 = 225 \]
Taking the square root:
\[ \text{adjacent} = \sqrt{225} = 15 \]

### Step 3: Calculate \( \cos \theta \)
The cosine of an angle in a right triangle is given by:
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substituting the values we found:
\[ \cos \theta = \frac{15}{17} \]

### Step 4: Calculate \( \tan \theta \)
The tangent of an angle is given by:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
Substituting the values:
\[ \tan \theta = \frac{8}{15} \]

### Final Answers
Thus, the values are:
- \( \cos \theta = \frac{15}{17} \)
- \( \tan \theta = \frac{8}{15} \)

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