If `13 e^(i tan^(-1)(5//12))= a+ib`, then |a| + |b| is equal to ____________.

Answer

To solve the problem, we need to express \( 13 e^{i \tan^{-1}(\frac{5}{12})} \) in the form \( a + ib \) and then find \( |a| + |b| \).

### Step-by-Step Solution:

1. **Identify the expression**: We have the expression \( 13 e^{i \tan^{-1}(\frac{5}{12})} \).

2. **Use Euler's formula**: According to Euler's formula, we can express \( e^{i\theta} \) as \( \cos(\theta) + i\sin(\theta) \). Therefore,
\[ 13 e^{i \tan^{-1}(\frac{5}{12})} = 13 \left( \cos\left(\tan^{-1}\left(\frac{5}{12}\right)\right) + i \sin\left(\tan^{-1}\left(\frac{5}{12}\right)\right) \right) \]

3. **Determine \( \cos(\theta) \) and \( \sin(\theta) \)**: Let \( \theta = \tan^{-1}\left(\frac{5}{12}\right) \). This means that:
\[ \tan(\theta) = \frac{5}{12} \]
We can visualize this with a right triangle where the opposite side is 5 and the adjacent side is 12. The hypotenuse can be calculated using the Pythagorean theorem:
\[ \text{hypotenuse} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

4. **Calculate \( \cos(\theta) \) and \( \sin(\theta) \)**:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13} \]
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13} \]

5. **Substitute back into the expression**:
\[ 13 e^{i \tan^{-1}(\frac{5}{12})} = 13 \left( \frac{12}{13} + i \frac{5}{13} \right) \]
Simplifying this gives:
\[ = 12 + 5i \]

6. **Identify \( a \) and \( b \)**: From the expression \( a + ib \), we have:
\[ a = 12 \quad \text{and} \quad b = 5 \]

7. **Calculate \( |a| + |b| \)**:
\[ |a| + |b| = |12| + |5| = 12 + 5 = 17 \]

### Final Answer:
Thus, \( |a| + |b| \) is equal to \( 17 \).

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