In trigonometry, many formulas are easier to derive than to memorize. Cosine of double angle is a wonderful formula! It allows you to obtain formulas for reducing degrees and formulas for half angles.
So, we need the cosine of the double angle and the trigonometric unit:
They are even similar: in the double angle cosine formula it is the difference between the squares of the cosine and sine, and in the trigonometric unit it is their sum. If we express the cosine from the trigonometric unit:
and substitute it into the cosine of the double angle, we get:
This is another double angle cosine formula:
This formula is the key to obtaining the reduction formula:
So, the formula for reducing the degree of sine is:
If in it the alpha angle is replaced by a half angle alpha in half, and the double angle two alpha is replaced by an alpha angle, then we obtain the half angle formula for sine:
Now we can express the sine from the trigonometric unit:
Let's substitute this expression into the double angle cosine formula:
We got another formula for the cosine of a double angle:
This formula is the key to finding the formula for reducing the power of cosine and the half angle for cosine.
Thus, the formula for reducing the degree of cosine is:
If we replace α with α/2, and 2α with α, we obtain the formula for the half argument for the cosine:
Since tangent is the ratio of sine to cosine, the formula for tangent is:
Cotangent is the ratio of cosine to sine. Therefore, the formula for cotangent is:
Of course, in the process of simplifying trigonometric expressions, there is no point in deriving the formula for half an angle or reducing a degree every time. It is much easier to put a sheet of paper with formulas in front of you. And simplification will move faster, and visual memory will turn on memorization.
But it’s still worth deriving these formulas several times. Then you will be absolutely sure that during the exam, when it is not possible to use a cheat sheet, you will easily get them if the need arises.
Now we will look at the question of how to plot trigonometric functions of multiple angles ωx, Where ω - some positive number.
To graph a function y = sin ωx Let's compare this function with the function we have already studied y = sin x. Let's assume that when x = x 0 function y = sin x takes the value equal to 0. Then
y 0 = sin x 0 .
Let us transform this relationship as follows:
Therefore, the function y = sin ωx at X = x 0 / ω takes the same value at 0 , which is the same as the function y = sin x at x = x 0 . This means that the function y = sin ωx repeats its meanings in ω times more often than the function y = sin x. Therefore, the graph of the function y = sin ωx obtained by "compressing" the graph of the function y = sin x V ω times along the x axis.
For example, the graph of a function y = sin 2x obtained by “compressing” a sinusoid y = sin x twice along the abscissa axis.
Graph of a function y = sin x / 2 is obtained by “stretching” the sinusoid y = sin x twice (or “compressing” it by 1 / 2 times) along the x axis.
Since the function y = sin ωx repeats its meanings in ω
times more often than the function
y = sin x, then its period is ω
times less than the period of the function y = sin x. For example, the period of the function y = sin 2x equals 2π/2 = π
, and the period of the function y = sin x / 2
equals π
/
x/ 2
= 4π .
It is interesting to study the behavior of the function y = sin ax using the example of animation, which can be very easily created in the program Maple:
Graphs of other trigonometric functions of multiple angles are constructed in a similar way. The figure shows the graph of the function y = cos 2x, which is obtained by “compressing” the cosine wave y = cos x twice along the x-axis.
Graph of a function y = cos x / 2 obtained by “stretching” the cosine wave y = cos x doubled along the x axis.
In the figure you see the graph of the function y = tan 2x, obtained by “compressing” the tangentsoids y = tan x twice along the abscissa axis.
Graph of a function y = tg x/ 2 , obtained by “stretching” the tangentsoids y = tan x doubled along the x axis.
And finally, the animation performed by the program Maple:
Exercises
1. Construct graphs of these functions and indicate the coordinates of the points of intersection of these graphs with the coordinate axes. Determine the periods of these functions.
A). y = sin 4x/ 3 G). y = tan 5x/ 6 and). y = cos 2x/ 3
b). y=cos 5x/ 3 d). y = ctg 5x/ 3 h). y=ctg x/ 3
V). y = tan 4x/ 3 e). y = sin 2x/ 3
2. Determine the periods of functions y = sin (πх) And y = tg (πх/2).
3. Give two examples of functions that take all values from -1 to +1 (including these two numbers) and change periodically with period 10.
4 *. Give two examples of functions that take all values from 0 to 1 (including these two numbers) and change periodically with a period π/2.
5. Give two examples of functions that take all real values and vary periodically with period 1.
6 *. Give two examples of functions that accept all negative values and zero, but do not accept positive values, and change periodically with a period of 5.
The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are specified trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this explains the abundance of trigonometric formulas. Some formulas connect trigonometric functions of the same angle, others - functions of a multiple angle, others - allow you to reduce the degree, fourth - express all functions through the tangent of a half angle, etc.
In this article we will list in order all the basic trigonometric formulas, which are sufficient to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them by purpose and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities define the relationship between sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function in terms of any other.
For a detailed description of these trigonometry formulas, their derivation and examples of application, see the article.
Reduction formulas
Reduction formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, as well as the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.
The rationale for these formulas, a mnemonic rule for memorizing them and examples of their application can be studied in the article.
Addition formulas
Trigonometric addition formulas show how trigonometric functions of the sum or difference of two angles are expressed in terms of trigonometric functions of those angles. These formulas serve as the basis for deriving the following trigonometric formulas.
Formulas for double, triple, etc. angle
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. angle
Half angle formulas
Half angle formulas show how trigonometric functions of a half angle are expressed in terms of the cosine of a whole angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric formulas for reducing degrees are designed to facilitate the transition from natural powers of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow you to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
Main purpose formulas for the sum and difference of trigonometric functions is to go to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow you to factor the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to a sum or difference is carried out using the formulas for the product of sines, cosines and sine by cosine.
Universal trigonometric substitution
We complete our review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement was called universal trigonometric substitution. Its convenience lies in the fact that all trigonometric functions are expressed in terms of the tangent of a half angle rationally without roots.
References.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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