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!