Now we will consider the question of how to build graphs trigonometric functions multiple angles ωx, Where ω is some positive number.
To plot a function y = sin ωx Let's compare this function with the function we have already studied y = sinx. Let's assume that at x = x 0 function y = sin x takes a value equal to 0 . Then
y 0 = sin x 0 .
Let's transform this ratio as follows:
Therefore, the function y = sin ωx at X = x 0 / ω takes the same value at 0 , which is the function y = sin x at x = x 0 . And this means that the function y = sin ωx repeats its values in ω times more often than the function y = sin x. So the graph of the function y = sin ωx obtained by "compressing" the graph of the function y = sinx V ω times along the x-axis.
For example, the graph of the function y \u003d sin 2x obtained by "compressing" the sinusoid y = sinx twice along the abscissa.
Function Graph y \u003d sin x / 2 obtained by "stretching" the sinusoid y \u003d sin x twice (or "compressing" in 1 / 2 times) along the x-axis.
Since the function y = sin ωx repeats its values in ω
times more often than the function
y = sinx, then its period in ω
times less than the period of the function y = sinx. For example, the period of the function y \u003d sin 2x equals 2π / 2 = π
, and the period of the function y \u003d sin x / 2
equals π
/
x / 2
= 4π .
It is interesting to study the behavior of the function y \u003d sin ax on the example of animation, which can be very easily created in the program maple:
Similarly, graphs are constructed for other trigonometric functions of multiple angles. The figure shows a graph of the function y = cos 2x, which is obtained by "compressing" the cosine y = cos x twice along the x-axis.
Function Graph y = cos x / 2 obtained by "stretching" the cosine wave y = cos x twice along the x-axis.
In the figure you see a graph of the function y = tg 2x, obtained by "compressing" the tangentoid y = tg x twice along the abscissa.
Function Graph y = tg x / 2 , obtained by "stretching" the tangentoid y = tg x twice along the x-axis.
And finally, the animation performed by the program maple:
Exercises
1. Build 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=tg 5x / 6 and). y = cos 2x / 3
b). y= cos 5x / 3 e). y=ctg 5x / 3 h). y=ctg x / 3
V). y=tg 4x / 3 e). y = sin 2x / 3
2. Define Function Periods y \u003d sin (πx) And y = tg (πх / 2).
3. Give two examples of a function that takes all values from -1 to +1 (including these two numbers) and changes periodically with a period of 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 change 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.
In trigonometry, many formulas are easier to deduce than to memorize. The cosine of a double angle is a wonderful formula! It allows you to get the reduction formulas and half angle formulas.
So, we need the cosine of the double angle and the trigonometric unit:
They are even similar: in the formula of the cosine of a double angle - the difference between the squares of the cosine and sine, and in the trigonometric unit - 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 formula for the cosine of a double angle:
This formula is the key to getting the reduction formula:
So, the formula for lowering the degree of the sine is:
If in it the angle alpha is replaced by a half angle alpha in half, and the double angle two alpha is replaced by the angle alpha, then we get the formula for the half angle for the sine:
Now, from the trigonometric unit, we express the sine:
Substitute this expression into the formula for the cosine of a double angle:
We got another formula for the cosine of a double angle:
This formula is the key to finding the cosine reduction and half angle formula for cosine.
Thus, the formula for lowering the degree of cosine is:
If we replace α by α/2 in it, and 2α by α, then we get 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. So the formula for the cotangent is:
Of course, in the process of simplifying trigonometric expressions, there is no point in deriving half-angle formulas or lowering the degree every time. It is much easier to put a sheet of formulas in front of you. And simplification will advance faster, and visual memory will turn on for memorization.
But it is still worth deriving these formulas several times. Then you will be absolutely sure that during the exam, when there is no way to use a cheat sheet, you can easily get them if the need arises.
The ratios between the main trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this also explains the abundance of trigonometric formulas. Some formulas connect the trigonometric functions of the same angle, others - the functions of a multiple angle, others - allow you to lower the degree, the fourth - to express all functions through the tangent of a half angle, etc.
In this article, we list in order all the basic trigonometric formulas, which are enough to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them according to their purpose, and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities set the relationship between the 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 through any other.
For a detailed description of these trigonometry formulas, their derivation and application examples, see the article.
Cast formulas
Cast 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, and also the property of shift by a given angle. These trigonometric formulas allow you to work with arbitrary angles move on to work 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 the trigonometric functions of the sum or difference of two angles are expressed in terms of the trigonometric functions of these angles. These formulas serve as the basis for the derivation of 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 the 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 the trigonometric functions of a half angle are expressed in terms of the cosine of an integer angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Reduction formulas
Trigonometric formulas for decreasing 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 one to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
The main purpose sum and difference formulas for trigonometric functions consists in the transition 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 factoring 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 the sum or difference is carried out through the formulas for the product of sines, cosines and sine by cosine.
Universal trigonometric substitution
We complete the review of the basic formulas of trigonometry with formulas expressing trigonometric functions in terms of the tangent of a half angle. This replacement is 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.
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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