The approximate value of `sin (60^(@) 45')," if "1^(@)=0.0175^(@)` is

Options

A. `0.8752`
B. `0.8762`
C. `0.8725`
D. `0.8726`

Answer

To find the approximate value of \( \sin(60^\circ 45') \), we can follow these steps:

### Step 1: Convert the angle into degrees
The angle \( 60^\circ 45' \) can be expressed in degrees as:
\[ 60^\circ + 45' = 60^\circ + \frac{45}{60}^\circ = 60^\circ + 0.75^\circ = 60.75^\circ \]

**Hint:** Remember that 1 minute is \( \frac{1}{60} \) of a degree.

### Step 2: Define \( x \) and \( \Delta x \)
Let:
\[ x = 60^\circ \quad \text{and} \quad \Delta x = 0.75^\circ \]

### Step 3: Convert degrees to radians
Given that \( 1^\circ = 0.0175 \) radians, we convert \( \Delta x \) into radians:
\[ \Delta x = 0.75^\circ \times 0.0175 \, \text{radians/degree} = 0.013125 \, \text{radians} \]

**Hint:** Use the conversion factor provided to change degrees to radians.

### Step 4: Differentiate \( y = \sin x \)
The derivative of \( y = \sin x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \cos x \]

### Step 5: Evaluate \( \cos x \) at \( x = 60^\circ \)
We know:
\[ \cos(60^\circ) = \frac{1}{2} \]

### Step 6: Calculate \( \Delta y \)
Using the formula for the change in \( y \):
\[ \Delta y = \frac{dy}{dx} \cdot \Delta x = \cos(60^\circ) \cdot \Delta x = \frac{1}{2} \cdot 0.013125 = 0.0065625 \]

**Hint:** Make sure to multiply the derivative by the change in angle in radians.

### Step 7: Find \( \sin(60^\circ) \)
We know:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866025 \]

### Step 8: Calculate \( \sin(60^\circ 45') \)
Now, we can approximate:
\[ \sin(60^\circ 45') \approx \sin(60^\circ) + \Delta y \approx 0.866025 + 0.0065625 \approx 0.8725875 \]

### Step 9: Round to the nearest option
Rounding \( 0.8725875 \) gives us approximately \( 0.8726 \).

**Final Answer:**
The approximate value of \( \sin(60^\circ 45') \) is \( 0.8726 \).