If `costheta=(5)/(13)`, then find the value of `tantheta`.

Options

A. `(5)/(12)`
B. `(12)/(13)`
C. `(12)/(5)`
D. `(13)/(5)`

Explanation

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

### Step 1: Understand the relationship between sine, cosine, and tangent
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus, to find \( \tan \theta \), we need to find \( \sin \theta \).

### Step 2: Use the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta = \frac{5}{13} \) into the equation:
\[ \sin^2 \theta + \left(\frac{5}{13}\right)^2 = 1 \]

### Step 3: Calculate \( \cos^2 \theta \)
Calculating \( \left(\frac{5}{13}\right)^2 \):
\[ \left(\frac{5}{13}\right)^2 = \frac{25}{169} \]
Now substitute this back into the Pythagorean identity:
\[ \sin^2 \theta + \frac{25}{169} = 1 \]

### Step 4: Solve for \( \sin^2 \theta \)
To isolate \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \frac{25}{169} \]
Convert 1 to a fraction with a denominator of 169:
\[ 1 = \frac{169}{169} \]
Now perform the subtraction:
\[ \sin^2 \theta = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \]

### Step 5: Find \( \sin \theta \)
Taking the square root of both sides gives:
\[ \sin \theta = \sqrt{\frac{144}{169}} = \frac{12}{13} \]
(Note: We take the positive root since we are considering angles in the first quadrant.)

### Step 6: Calculate \( \tan \theta \)
Now that we have both \( \sin \theta \) and \( \cos \theta \), we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{12}{5} \]

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