Sideway
output.to from Sideway
Draft for Information Only

Content

Derivatives of Trigonometric Functions
  Limits and Trigonometric Functions
   Limits and Sine Function
   Limits and Cosine Function
   Plots of Limits and Trigonometric Functions
  Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions

Trigonometric functions are very important in practical application. Trigonmetric function f is continuous when it is defined at every real numbers x in the domain. To determine the derivatives, the real number x is the angle of the trigonometric function and is measured in radian.

Limits and Trigonometric Functions

Trigonometric functions are functions of an angle. The three basic are sine cosine and tangent. The geometric definition of trigonometric functions can be related by sides of a triangle. One important limit related to finding derivatives of trigonometric functions is the ratio of value of a trigonometric function to the angle of the trigonometric function. Geometrically, the trigonometric functions can represented by constructing triangles in a unit circle with radius r=1.

 IMAGE...

Limits and Sine Function

According to the graph, consider the angle x in radian between 0 and π/2  when x approaching zero and not equal to zero, the relationship of the area of triangle w/sin x, the area of circular sector w/arc x, and the area of triangle w/tan x are:

 IMAGE...

As x decrease, cos x  will increase. (sin x)/x will increase also but less than 1. Imply as x approach +0, limit of (sin x)/x exists. Rearrange the inequality:

 IMAGE...

As x approach +0, i.e. δ>x>0 imply:

 IMAGE...

Similarly as x approach -0, imply:

 IMAGE...

Therefore:

 IMAGE...

Limits and Cosine Function

Another important limit is:

 IMAGE...

Since cosine function and sine function are related by a right angle triangle and both functions are continuous in every numbe of the domain, the cosine function can be transformed through algebraic manipulations:

 IMAGE...

The limit of the function can be determined by:

 IMAGE...

Plots of Limits and Trigonometric Functions

The inequality, limits and trigonometric functions can be plotted as in the following figure:

 IMAGE...

Derivatives of Trigonometric Functions

  1. Derivative of Sine Function

     IMAGE...

    Proof:

     IMAGE...
  2. Derivative of Cosine Function

     IMAGE...

    Proof:

     IMAGE...
  3. Derivative of Tangent Function

     IMAGE...

    Proof:

     IMAGE...
  4. Derivative of Cotangent Function

     IMAGE...

    Proof:

     IMAGE...
  5. Derivative of Secant Function

     IMAGE...

    Proof:

     IMAGE...
  6. Derivative of Cosecant Function

     IMAGE...

    Proof:

     IMAGE...

©sideway

ID: 110900006 Last Updated: 9/12/2011 Revision: 0 Ref:

close

References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
close

Latest Updated LinksValid XHTML 1.0 Transitional Valid CSS!Nu Html Checker Firefox53 Chromena IExplorerna
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 339

Reference 79

Computer

Hardware 249

Software

Application 213

Digitization 32

Latex 52

Manim 205

KB 1

Numeric 19

Programming

Web 289

Unicode 504

HTML 66

CSS 65

SVG 46

ASP.NET 270

OS 429

DeskTop 7

Python 72

Knowledge

Mathematics

Formulas 8

Algebra 84

Number Theory 206

Trigonometry 31

Geometry 34

Coordinate Geometry 2

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Natural Sciences

Matter 1

Electric 27

Biology 1

Geography 1


Copyright © 2000-2024 Sideway . All rights reserved Disclaimers last modified on 06 September 2019