Q. 73 - Calculus

Question

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$as alternating series. Then use Theorem $7.38$ to approximate the value of the specified quantities to within$0.001$of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$of its value? In Exercises $73-76$be sure to convert to radian measure first.

$\sin 2°$

Step-by-Step Solution

Answer

The approvimate the value of $\sin 2°$  up to three decimal places is $0.0349$

Also, to approximate the quantity $\sin 2°$to within ${10}^{-4}$ of its value, the number of terms required is $2$

step 1: Given information

Consider the function$\sin 2°$

Step 2: Calculation

Let us first convert the degree into radian measure.

Since$\pi radians=180°,$ therefore$1°=\frac{\pi }{180}radian$

So,

$2°=\frac{2\mathrm{\pi }}{180}$

$=\frac{\mathrm{\pi }}{90}radian$

Also, let us consider$f\left(x\right)=\sin x$

The Maclaurin series for the function $f\left(x\right)=\sin x$is

$f\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{(2k+1)!}{x}^{2k+1}$

So, to find the Maclaurin series for the function $\sin \frac{\pi }{90}$, put $x=\frac{\pi }{90}$Therefore,

$f\left(x=\frac{\pi }{90}\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{(2k+1)!}{\left(\frac{\pi }{90}\right)}^{2k+1}$

That is

$\sin \frac{\pi }{90}=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{(2k+1)!}{\left(\frac{\pi }{90}\right)}^{2k+1}$

Now, to approximate the value of $\sin \frac{\pi }{90}$ up to three decimal places, let us first write its corresponding Maclaurin series in expanded form.

So,

$$\begin{gathered} \sin \frac{\pi }{90}=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{(2k+1)!}{\left(\frac{\pi }{90}\right)}^{2k+1} \\ =\frac{\pi }{90}-\frac{1}{3!}{\left(\frac{\pi }{90}\right)}^3+\frac{1}{5!}{\left(\frac{\pi }{90}\right)}^5-\frac{1}{7!}{\left(\frac{\pi }{90}\right)}^7+\frac{1}{9!}{\left(\frac{\pi }{90}\right)}^9-\frac{1}{11!}{\left(\frac{\pi }{90}\right)}^{11} \\ =0.0349-0.00000708 \end{gathered}$$

Therefore, the approximate the value of $\sin 2°$ up to three decimal places is $0.0349$

Also, to approximate the quantity $\sin 2°$ to within ${10}^{-6}$ of its value, the number of terms required is $2$