Write the radian measure of `5^(@) 37' 30"`

Answer

To convert the angle \(5^\circ 37' 30''\) into radians, we will follow these steps:

### Step 1: Convert seconds to minutes
1. We know that \(1 \text{ minute} = 60 \text{ seconds}\).
2. Therefore, \(30''\) (seconds) can be converted to minutes:
\[ 30'' = \frac{30}{60} = 0.5' \]

### Step 2: Add the minutes
3. Now, we can add this to the minutes:
\[ 5^\circ 37' 30'' = 5^\circ 37' + 0.5' = 5^\circ 37.5' \]

### Step 3: Convert minutes to degrees
4. Next, we convert the total minutes into degrees. We know that \(1 \text{ degree} = 60 \text{ minutes}\):
\[ 37.5' = \frac{37.5}{60} = 0.625^\circ \]
5. Therefore, we can express the angle in degrees:
\[ 5^\circ + 0.625^\circ = 5.625^\circ \]

### Step 4: Convert degrees to radians
6. To convert degrees to radians, we use the conversion factor \(\frac{\pi \text{ radians}}{180^\circ}\):
\[ \text{Radians} = 5.625^\circ \times \frac{\pi}{180} \]

### Step 5: Simplify the expression
7. Now, we calculate:
\[ \text{Radians} = \frac{5.625 \pi}{180} \]
To simplify \(5.625\):
\[ 5.625 = \frac{5625}{1000} = \frac{225}{40} = \frac{45}{8} \]
Thus,
\[ \text{Radians} = \frac{45 \pi}{8 \times 180} = \frac{45 \pi}{1440} \]

### Step 6: Further simplification
8. Simplifying \(\frac{45}{1440}\):
\[ \frac{45}{1440} = \frac{1}{32} \]
Therefore,
\[ \text{Radians} = \frac{\pi}{32} \]

### Final Answer
The radian measure of \(5^\circ 37' 30''\) is:
\[ \frac{\pi}{32} \text{ radians} \]

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