QQuestionMathematics
QuestionMathematics
The $4^{\text {th }}$ degree Taylor polynomial for $\sin (x)$ centered at $a=\frac{\pi}{6}$ is given by the following.
\sin (x)=\frac{1}{2}+\frac{\sqrt{3}}{2}\left(x-\frac{\pi}{6}\right)-\frac{1}{4}\left(x-\frac{\pi}{6}\right)^{2}-\frac{\sqrt{3}}{12}\left(x-\frac{\pi}{6}\right)^{3}+\frac{1}{48}\left(x-\frac{\pi}{6}\right)^{4}+R_{4}(x)
Using this, estimate $\sin \left(40^{\circ}\right)$ correct to five decimal places.
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Answer
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Step 1:: Convert the given angle from degrees to radians.
We know that $40^{\circ} = \frac{40 \times \pi}{180} \text{ radians}$.
So, substitute this value into the Taylor polynomial.
Step 2:: Substitute the given value into the Taylor polynomial.
\begin{align*} \end{align*}
Step 3:: Calculate the approximate value.
After calculating the values, we get: \sin \left(40^{\circ}\right) \approx 0.64279
Step 4:: Round the result to five decimal places.
\sin \left(40^{\circ}\right) \approx 0.64279
Final Answer
The estimated value of $\sin \left(40^{\circ}\right)$ correct to five decimal places is approximately $0.64279$.
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