Question

Use the Taylor series to estimate LaTeX: sin(x)s i n ( x ) about LaTeX: a = 0a = 0. Truncate to 6 terms (including terms that simplify to 0). What is the relative error of using this truncated Taylor series to compute the value of LaTeX: sin(\frac{\pi}{4})s i n ( π 4 )?

Answer 1

Answer:

$E_{r}=0.005%$ %

Step-by-step explanation:

The Taylor series of a function around a value ("a" is this case) is:

$f(x)=f(a)+\frac{f'(a)(x-a)}{1!}+\frac{f''(a)(x-a)^{2}}{2!}+...$

• f' is the first derivative, f'' the second, and so on
• a=0

If f(x)=sin(x), the Taylor series will be:

$sin(x)=sin(0)+\frac{cos(0)(x-0)}{1!}-\frac{sin(0)(x-0)^{2}}{2!}-\frac{cos(0)(x-0)^{3}}{3!}+\frac{sin(0)(x-0)^{4}}{4!}+\frac{cos(0)(x-0)^{5}}{5!}$

We truncate to 6 term as the question says.

Now, sin(0)=0 and cos(0)=1.

$sin(x)=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$

Let's find sin(π/4) using the Taylor series and with the calculator, and then calculate the relative error.

Taylor series

$sin(\frac{\pi}{4})=\frac{\pi}{4}-\frac{(\frac{\pi}{4})^{3}}{6}+\frac{(\frac{\pi}{4})^{5}}{120}=0.707143058$

Calculator

$sin(\frac{\pi}{4})=0.7071067812$

The relative error will be:            

$E_{r}=\left |1-\frac{V_{aprox}}{V_{real}} \right |*100=0.005\%$  

I hope it helps you!