A point has rectangular coordinates (3,4) . The polar coordinates are `(5 , theta)` . What is the value of `theta` ?

Options

A. `30^@`
B. `37^@`
C. `51^@`
D. `53^@`

Explanation

To find the value of \( \theta \) for the polar coordinates of the point with rectangular coordinates \( (3, 4) \), we can use the relationship between rectangular and polar coordinates. The formulas are:

1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)

Given:
- Rectangular coordinates: \( (x, y) = (3, 4) \)
- Polar coordinates: \( (r, \theta) = (5, \theta) \)

### Step 1: Use the formula for \( x \)

From the first formula, we have:
\[ x = r \cos \theta \]
Substituting the known values:
\[ 3 = 5 \cos \theta \]

### Step 2: Solve for \( \cos \theta \)

To isolate \( \cos \theta \), we divide both sides by 5:
\[ \cos \theta = \frac{3}{5} \]

### Step 3: Use the formula for \( y \)

Now, using the second formula:
\[ y = r \sin \theta \]
Substituting the known values:
\[ 4 = 5 \sin \theta \]

### Step 4: Solve for \( \sin \theta \)

To isolate \( \sin \theta \), we divide both sides by 5:
\[ \sin \theta = \frac{4}{5} \]

### Step 5: Find \( \theta \)

Now that we have both \( \cos \theta \) and \( \sin \theta \), we can find \( \theta \) using the inverse trigonometric functions. We can use the tangent function as well:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

Now, we find \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Result

Thus, the value of \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]