Chapter 1: Q39P (page 32)

$f (x) = s i n x, a = π / 2$

Short Answer

Expert verified

$\sin (x) = 1 - \frac{{(x - \frac{π}{2})}^{2}}{2} + \frac{x {(x - \frac{π}{2})}^{4}}{24} - \frac{{(x - \frac{π}{2})}^{6}}{720} + \dots$

Step by step solution

Given information.

The Maclaurin expansion of the function $f (x) = s i n x, a = π / 2$ is to be evaluated.

Definition of Maclaurin series.

The definition of the Maclaurin series is defined.

$c o s (x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots$

Begin by stating the Maclaurin series.

State the Maclaurin series.

$\sin (x) = \sin [\frac{π}{2} + (x - \frac{π}{2})] = x - \frac{x^{3}}{6} + \frac{x^{3}}{120} + \dots$

Use the fact $\sin (x + \frac{π}{2}) = \cos (x)$ to equate the previous equation.

$\sin [\frac{π}{2} + (x - \frac{π}{2})] = \cos (x - \frac{π}{2})$

$\cos (x) = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \dots$

Make appropriate substitution in the Maclaurin series and evaluate.

Replace x with the expression $(x - \frac{π}{2})$ in the third defined Maclaurin series.

$\sin (x) = \cos (x - \frac{π}{2})$

localid="1657384732250" $\sin (x) = 1 - \frac{{(x - \frac{π}{2})}^{2}}{2} + \frac{x {(x - \frac{π}{2})}^{4}}{24} - \frac{{(x - \frac{π}{2})}^{6}}{720} + \dots$

Thus, the final answer islocalid="1657384971196" $\sin (x) = 1 - \frac{{(x - \frac{π}{2})}^{2}}{2} + \frac{x {(x - \frac{π}{2})}^{4}}{24} - \frac{{(x - \frac{π}{2})}^{6}}{720} + \dots$

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$f (x) = s i n x, a = π / 2$

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Begin by stating the Maclaurin series.

Make appropriate substitution in the Maclaurin series and evaluate.

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f(x)=sin x, a=π/2

Short Answer

Step by step solution

Given information.

Definition of Maclaurin series.

Begin by stating the Maclaurin series.

Make appropriate substitution in the Maclaurin series and evaluate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Dynamics

Work Energy and Power

Waves Physics

Famous Physicists

Nuclear Physics

Study anywhere. Anytime. Across all devices.

$f (x) = s i n x, a = π / 2$