If sin `theta=(5)/(13)` , then find cos `theta`

Answer

To find \( \cos \theta \) given that \( \sin \theta = \frac{5}{13} \), we can follow these steps:

### Step 1: Understand the relationship between sine and cosine
We know that:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
From the given information, we can identify:
- Opposite side (perpendicular) = 5
- Hypotenuse = 13

### Step 2: Use the Pythagorean theorem to find the base
According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Let the adjacent side (base) be \( b \). Thus, we can write:
\[ 13^2 = 5^2 + b^2 \]
Calculating the squares:
\[ 169 = 25 + b^2 \]

### Step 3: Solve for the base
Now, we can rearrange the equation to solve for \( b^2 \):
\[ b^2 = 169 - 25 \]
\[ b^2 = 144 \]
Taking the square root of both sides gives us:
\[ b = \sqrt{144} = 12 \]

### Step 4: Find \( \cos \theta \)
Now that we have the base (adjacent side), we can find \( \cos \theta \):
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{\text{hypotenuse}} = \frac{12}{13} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{12}{13} \]

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