Convert `40^(@)21'` into radian measure.

Answer

To convert \(40^\circ 21'\) into radian measure, we will follow these steps:

### Step 1: Convert minutes to degrees
We know that:
1 degree = 60 minutes

Thus, to convert 21 minutes into degrees:
\[ 21' = \frac{21}{60} \text{ degrees} \]

### Step 2: Add degrees and converted minutes
Now, we can express \(40^\circ 21'\) as:
\[ 40^\circ + 21' = 40^\circ + \frac{21}{60}^\circ \]

### Step 3: Find a common denominator
To add these two terms, we can express \(40^\circ\) in terms of a fraction:
\[ 40^\circ = \frac{40 \times 60}{60} = \frac{2400}{60}^\circ \]
Now, we can add:
\[ \frac{2400}{60}^\circ + \frac{21}{60}^\circ = \frac{2400 + 21}{60}^\circ = \frac{2421}{60}^\circ \]

### Step 4: Convert degrees to radians
We know that:
\[ 180^\circ = \pi \text{ radians} \]
To convert degrees to radians, we use the conversion factor:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Thus, we can convert \(\frac{2421}{60}^\circ\) to radians:
\[ \text{radians} = \frac{2421}{60} \times \frac{\pi}{180} \]

### Step 5: Simplify the expression
Now, we simplify:
\[ \text{radians} = \frac{2421 \pi}{60 \times 180} \]
Calculating \(60 \times 180\):
\[ 60 \times 180 = 10800 \]
So we have:
\[ \text{radians} = \frac{2421 \pi}{10800} \]

### Step 6: Final answer
Thus, the final answer in radian measure is:
\[ \text{radians} = \frac{2421 \pi}{10800} \]

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