The approximate value of `sin (60^(@) 0' 10'')," if "1^(@)=0.175^(@)` is

Options

A. `0.8660243`
B. `0.8660244`
C. `0.866243`
D. `0.866244`

Answer

To find the approximate value of \( \sin(60^\circ 0' 10'') \) given that \( 1^\circ = 0.175 \) radians, we can follow these steps:

### Step 1: Convert the angle to radians
The angle \( 60^\circ 0' 10'' \) can be converted to degrees first:
- \( 10'' = \frac{10}{3600} \) degrees (since there are 3600 seconds in a degree).
- Therefore, \( 60^\circ 0' 10'' = 60 + \frac{10}{3600} = 60 + \frac{1}{360} \) degrees.

### Step 2: Calculate the total degrees
Now, we can calculate:
\[ 60 + \frac{1}{360} \approx 60.0027778^\circ \]

### Step 3: Convert degrees to radians
Using the conversion \( 1^\circ = 0.0175 \) radians:
\[ \text{Total radians} = (60.0027778) \times 0.0175 \]

### Step 4: Calculate the sine using the approximation
Let \( x = 60^\circ \) and \( h = 10'' \). We can use the formula for small angles:
\[ \sin(x + h) \approx \sin x + h \cos x \]
Where:
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \cos(60^\circ) = \frac{1}{2} \)

### Step 5: Calculate \( h \)
Convert \( h \) (10 seconds) to radians:
\[ h = 10'' = \frac{10}{3600} \text{ degrees} = \frac{10 \times 0.0175}{3600} \text{ radians} \]
Calculating this gives:
\[ h \approx \frac{0.175}{360} \approx 0.0004861 \text{ radians} \]

### Step 6: Substitute the values into the sine approximation
Now substituting into the sine approximation:
\[ \sin(60^\circ 0' 10'') \approx \sin(60^\circ) + h \cos(60^\circ) \]
\[ = \frac{\sqrt{3}}{2} + 0.0004861 \cdot \frac{1}{2} \]
\[ = 0.866025 + 0.00024305 \]
\[ \approx 0.866268 \]

### Final Result
Thus, the approximate value of \( \sin(60^\circ 0' 10'') \) is approximately \( 0.866268 \).

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