Question Statement

Find the approximate value of $\sin 31^{\circ }$ using differentials.

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Background and Explanation

These problems use linear approximation (differential approximation) to estimate function values near a known point using the tangent line. The method requires the formula $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$, where $x$ is chosen as a convenient value where the function and its derivative are easily calculated.

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Solution

Approximating $\sin 31^{\circ }$

To find the approximate value, we express the angle as a sum of a standard angle (where values are known) and a small increment: $\sin 31^{\circ }=\sin (30^{\circ }+1^{\circ })$

We choose $x=30^{\circ }$ (where $\sin 30^{\circ }=0.5$ is known) and the increment $\Delta x=1^{\circ }$.

Step 1: Convert the increment to radians For trigonometric derivatives to be valid, we must work in radians rather than degrees. We convert $dx=1^{\circ }$ to radians: $dx=1^{\circ }=\frac{\pi }{180}\text{ radians}\approx 0.01745\text{ rad}$

Step 2: Define the function and its derivative Let the function be: $f(x)=\sin x$ Differentiating with respect to $x$ gives: $f^{\prime }(x)=\cos x$

Step 3: Apply the linear approximation formula Using the linear approximation formula: $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\cdot dx$ This yields: $\sin (x+\Delta x)\approx \sin x+\cos x\cdot dx$

Step 4: Substitute the known values Substituting $x=30^{\circ }$ and $dx\approx 0.01745$: $\begin{array}{rl}\sin (30^{\circ }+1^{\circ })& \approx \sin 30^{\circ }+\cos 30^{\circ }(0.01745)\\ \sin 31^{\circ }& \approx 0.5+(0.866)(0.01745)\\ \sin 31^{\circ }& \approx 0.5+0.0151117\\ \sin 31^{\circ }& \approx 0.5151\end{array}$

Hence, the approximate value is: $\sin 31^{\circ }\approx 0.5151$

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Key Formulas or Methods Used

• Linear Approximation Formula: $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$ (Note: used for general differential problems)
• Derivative of Sine: $\frac{d}{dx}(\sin x)=\cos x$
• Degree to Radian Conversion: $1^{\circ }=\frac{\pi }{180}$ radians $\approx 0.01745$ rad

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Summary of Steps

1. Express the angle as $x+\Delta x$, choosing $x=30^{\circ }$ and $\Delta x=1^{\circ }$.
2. Convert the degree increment $\Delta x$ to radians: $dx=\frac{\pi }{180}\approx 0.01745$.
3. Define the function $f(x)=\sin x$ and find its derivative $f^{\prime }(x)=\cos x$.
4. Apply the differential formula and substitute the values: $\sin 31^{\circ }\approx 0.5+0.866(0.01745)\approx 0.5151$.