# Difference between revisions of "Sine-squared function"

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<math>\! \sin^2 x = x^2 - \frac{x^4}{3} + \frac{2x^6}{45} - \dots</math> | <math>\! \sin^2 x = x^2 - \frac{x^4}{3} + \frac{2x^6}{45} - \dots</math> | ||

− | == | + | ==Limit computations== |

+ | ===Order of zero=== | ||

We get the following limit from the power series: | We get the following limit from the power series: | ||

<math>\! \lim_{x \to 0} \frac{\sin^2x}{x^2} = 1</math> | <math>\! \lim_{x \to 0} \frac{\sin^2x}{x^2} = 1</math> | ||

− | + | Thus, the order of the zero of <math>\sin^2</math> at zero is 2 and the residue is 1. | |

− | + | This limit can be computed in many ways: | |

− | + | ||

+ | {| class="sortable" border="1" | ||

+ | ! Name of method for computing the limit !! Details | ||

+ | |- | ||

+ | | Simple manipulation, using <math>\lim_{x \to 0} (\sin x)/x = 1</math> || <math>\! \lim_{x \to 0} \frac{\sin^2x}{x^2} = \lim_{x \to 0} \left(\frac{\sin x}{x}\right)^2 = \left(\lim_{x \to 0} \frac{\sin x}{x}\right)^2 = 1^2 = 1</math> | ||

+ | |- | ||

+ | | Using the [[L'Hopital rule]] || <math>\lim_{x \to 0} \frac{\sin^2x}{x^2} \stackrel{*}{=} \lim_{x \to 0} \frac{\sin(2x)}{2x} \stackrel{*}{=} \frac{2\cos(2x)}{2} = 1</math> | ||

+ | |- | ||

+ | | Using the power series || We have <math>\sin^2x = x^2 - \frac{x^4}{3} + \dots</math>, so we get<br>\frac{\sin^2x}{x^2} = 1 - \frac{x^2}{3} + \dots</math>. Taking the limit as <math>x \to 0</math> gives 1. | ||

+ | |} | ||

+ | |||

+ | ===Higher order limits=== | ||

Further, we have, again from the power series: | Further, we have, again from the power series: | ||

<math>\! \lim_{x \to 0} \frac{x^2 - \sin^2x}{x^4} = \frac{1}{3}</math> | <math>\! \lim_{x \to 0} \frac{x^2 - \sin^2x}{x^4} = \frac{1}{3}</math> | ||

+ | |||

+ | This limit can be computed in many ways: | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Name of method for computing the limit !! Details | ||

+ | |- | ||

+ | | Using <math>\lim_{x \to 0} \frac{x - \sin x}{x^3} = 1/6</math> and <math>\lim_{x \to 0} \frac{\sin x}{x} = 1</math> || We have <math>\lim_{x \to 0} \frac{x^2 - \sin^2x}{x^4} = \lim{x \to 0} \frac{x - \sin x}{x^3} \cdot \frac{x + \sin x}{x} = \lim_{x \to 0} \frac{x - \sin x}{x^3} \cdot \lim_{x \to 0} \frac{x + \sin x}{x}</math>.<br>The first limit is <math>1/6</math> and the second limit is 2 from the given data. We thus get <math>(1/6) \cdot 2 = 1/3</math>. | ||

+ | |- | ||

+ | | Using the [[L'Hopital rule]] || <math>\lim_{x \to 0} \frac{x^2 - \sin^2x}{x^3} \stackrel{*}{=} \lim_{x \to 0} \frac{2x - \sin(2x)}{4x^3} \stackrel{*}{=} \lim_{x \to 0} \frac{2 - 2\cos(2x)}{12x^2} \stackrel{*}{=} \lim_{x \to 0} \frac{4\sin(2x)}{24x} \stackrel{*}{=} \lim_{x \to 0} \frac{8\cos(2x)}{24} = 1/3</math>. | ||

+ | |- | ||

+ | | Using the power series || We have <math>\sin^2x = x^2 - \frac{x^4}{3} + O(x^6)</math>, so <math>\frac{x^2 - \sin^2x}{x^4} = \frac{1}{3} - O(x^2)</math>, so the limit as <math>x \to 0</math> is 1/3. | ||

+ | |} |

## Revision as of 00:09, 4 September 2011

This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.

View a complete list of particular functions on this wiki

## Contents

## Definition

This function, denoted , is defined as the composite of the square function and the sine function. Explicitly, it is the map:

For brevity, we write as .

## Key data

Item | Value |
---|---|

Default domain | all real numbers, i.e., all of |

range | , i.e., absolute maximum value: 1, absolute minimum value: 0 |

period | , i.e., |

local maximum value and points of attainment | All local maximum values are equal to 1, and are attained at odd integer multiples of . |

local minimum value and points of attainment | All local minimum values are equal to 0, and are attained at integer multiples of . |

points of inflection (both coordinates) | odd multiples of , with value 1/2 at each point. |

derivative | , i.e., double-angle sine function. |

second derivative | |

derivative | times an expression that is or of , depending on the remainder of mod |

antiderivative | |

mean value over a period | 1/2 |

expression as a sinusoidal function plus a constant function | |

important symmetries | even function (follows from composite of even function with odd function is even, the square function being even, and the sine function being odd) more generally, miror symmetry about any vertical line of the form , an integer. Also, half turn symmetry about all points of the form . |

interval description based on increase/decrease and concave up/down | For each integer , the interval from to is subdivided into four pieces: : increasing and concave up : increasing and concave down : decreasing and concave down, : decreasing and concave up |

power series and Taylor series | The power series about 0 (which is hence also the Taylor series) is It is a globally convergent power series. |

## Graph

Here is the graph on the interval , drawn to scale:

Here is a close-up view of the graph between and . The dashed horizontal line indicates the mean value of :

The red dotted points indicate the points of inflection and the black dotted points indicate local extreme values.

Here is a picture showing the function (blue) and the cosine-squared function (purple) with the dashed line being . The picture illustrates that :

## Integration

### First antiderivative: using double angle formula

We use the identity:

We can now do the integration:

### First antiderivative: using integration by parts

We rewrite and use integration by parts in its recursive version:

We now rewrite and obtain:

Setting to be a choice of antiderivative so that the above holds without any freely floating constants, we get:

Rearranging, we get:

This gives:

So the general antiderivative is:

Using the double angle sine formula , we can verify that this matches with the preceding answer.

### Graph of function with antiderivative

In the picture below, we depict (blue) and the function (purple). This is the unique antiderivative that takes the value 0 at 0. The other antiderivatives can be obtained by vertically shifting the purple graph:

### Definite integrals

The part in the antiderivative signifies that the *linear* part of the antiderivative of has slope , and this is related to the fact that has a mean value of on any interval of length equal to the period. It is in fact clear that the function is a sinusoidal function about .

Thus, we have:

where is an integer.

### Higher antiderivatives

It is possible to antidifferentiate more than once. The antiderivative is the sum of a polynomial of degree and a trigonometric function with a period of .

## Power series and Taylor series

### Computation of power series

We can use the identity:

along with the power series for the cosine function, to find the power series for .

The power series for the cosine function converges to the function everywhere, and is:

The power series for is:

The power series for is:

Dividing by 2, we get the power series for :

Here's another formulation with the first few terms written more explicitly:

## Limit computations

### Order of zero

We get the following limit from the power series:

Thus, the order of the zero of at zero is 2 and the residue is 1.

This limit can be computed in many ways:

Name of method for computing the limit | Details |
---|---|

Simple manipulation, using | |

Using the L'Hopital rule | |

Using the power series | We have , so we get \frac{\sin^2x}{x^2} = 1 - \frac{x^2}{3} + \dots</math>. Taking the limit as gives 1. |

### Higher order limits

Further, we have, again from the power series:

This limit can be computed in many ways:

Name of method for computing the limit | Details |
---|---|

Using and | We have . The first limit is and the second limit is 2 from the given data. We thus get . |

Using the L'Hopital rule | . |

Using the power series | We have , so , so the limit as is 1/3. |