Find the complement of `37^(@)30`'

Answer

To find the complement of the angle \(37^\circ 30'\), we follow these steps:

### Step 1: Understand the concept of complementary angles
Complementary angles are two angles whose sum is \(90^\circ\). Therefore, to find the complement of an angle \(x\), we can use the formula:
\[ \text{Complement of } x = 90^\circ - x \]

### Step 2: Write down the angle
Given angle \(x = 37^\circ 30'\).

### Step 3: Convert \(90^\circ\) into minutes
Since \(1^\circ = 60'\), we can express \(90^\circ\) in terms of minutes:
\[ 90^\circ = 90 \times 60' = 5400' \]

### Step 4: Convert \(37^\circ 30'\) into minutes
Now, convert \(37^\circ 30'\) into minutes:
\[ 37^\circ 30' = 37 \times 60' + 30' = 2220' + 30' = 2250' \]

### Step 5: Subtract the angle from \(90^\circ\)
Now we can find the complement:
\[ \text{Complement} = 5400' - 2250' = 3150' \]

### Step 6: Convert the result back to degrees and minutes
Now, convert \(3150'\) back to degrees and minutes:
- First, find how many degrees are in \(3150'\):
\[ \text{Degrees} = \frac{3150'}{60} = 52^\circ \]
- Next, find the remaining minutes:
\[ \text{Remaining minutes} = 3150' - (52 \times 60') = 3150' - 3120' = 30' \]

### Final Answer
Thus, the complement of \(37^\circ 30'\) is:
\[ \text{Complement} = 52^\circ 30' \]

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