Use the differential to approximate the number sin(57◦
).

To approximate \( \sin(57^\circ) \) using differentials, we can proceed as follows:

### Step 1: Use a known angle close to \( 57^\circ \)
Let's choose a reference angle, \( 60^\circ \), which is close to \( 57^\circ \) and for which we know the exact value of the sine function:

\[
\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866
\]

### Step 2: Set up the differential formula
The approximation for \( \sin(57^\circ) \) can be done using the linear approximation formula:

\[
\Delta y \approx f'(x_0) \Delta x
\]

Where:
- \( f(x) = \sin(x) \)
- \( f'(x) = \cos(x) \) (since the derivative of sine is cosine)
- \( x_0 = 60^\circ \) (the known value)
- \( \Delta x = 57^\circ - 60^\circ = -3^\circ \)

First, convert \( 3^\circ \) to radians (since derivatives are best in radians):
\[
3^\circ = \frac{3 \pi}{180} \approx 0.0524 \, \text{radians}
\]

### Step 3: Apply the differential formula
Now we approximate \( \sin(57^\circ) \) using the differential:

\[
\sin(57^\circ) \approx \sin(60^\circ) + \cos(60^\circ) \cdot \Delta x
\]

We know that:
\[
\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866
\]
\[
\cos(60^\circ) = \frac{1}{2}
\]

Thus, we calculate:
\[
\sin(57^\circ) \approx 0.866 + \left( \frac{1}{2} \times (-0.0524) \right)
\]
\[
\sin(57^\circ) \approx 0.866 - 0.0262
\]
\[
\sin(57^\circ) \approx 0.8398
\]

### Final approximation
Therefore, using differentials, we approximate \( \sin(57^\circ) \) as \( \sin(57^\circ) \approx 0.8398 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you approximate \( \cos(35^\circ) \) using the same method?
2. How do you convert degrees to radians and vice versa?
3. Can you use the same differential method to approximate \( \tan(57^\circ) \)?
4. How would this method change if you used \( 55^\circ \) as a reference angle instead of \( 60^\circ \)?
5. Why are differentials used for approximation in calculus?

**Tip:** Always ensure you convert degrees to radians when applying calculus-based formulas, especially with trigonometric functions!

Use the differential to approximate the number sin(57°).

Learn how to use the differential method to approximate sin(57°). This example applies key trigonometric functions and the concept of linear approximation to estimate the value of sin(57°) using sin(60°) and cos(60°). Suitable for high school students in Grades 10-12.

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