This article is devoted to a detailed study trigonometric formulas ghosts Dan full list reduction formulas, examples of their use are shown, and proof of the correctness of the formulas is given. The article also provides a mnemonic rule that allows you to derive reduction formulas without memorizing each formula.
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Reduction formulas. List
Reduction formulas allow you to reduce basic trigonometric functions of angles of arbitrary magnitude to functions of angles lying in the range from 0 to 90 degrees (from 0 to π 2 radians). Operating with angles from 0 to 90 degrees is much more convenient than working with arbitrarily large values, which is why reduction formulas are widely used in solving trigonometry problems.
Before we write down the formulas themselves, let us clarify several important points for understanding.
- The arguments of trigonometric functions in reduction formulas are angles of the form ± α + 2 π · z, π 2 ± α + 2 π · z, 3 π 2 ± α + 2 π · z. Here z is any integer, and α is an arbitrary rotation angle.
- It is not necessary to learn all the reduction formulas, the number of which is quite impressive. There is a mnemonic rule that makes it easy to derive the desired formula. We will talk about the mnemonic rule later.
Now let's move directly to the reduction formulas.
Reduction formulas allow you to move from working with arbitrary and arbitrarily large angles to working with angles ranging from 0 to 90 degrees. Let's write all the formulas in table form.
Reduction formulas
sin α + 2 π z = sin α , cos α + 2 π z = cos α t g α + 2 π z = t g α , c t g α + 2 π z = c t g α sin - α + 2 π z = - sin α , cos - α + 2 π z = cos α t g - α + 2 π z = - t g α , c t g - α + 2 π z = - c t g α sin π 2 + α + 2 π z = cos α , cos π 2 + α + 2 π z = - sin α t g π 2 + α + 2 π z = - c t g α , c t g π 2 + α + 2 π z = - t g α sin π 2 - α + 2 π z = cos α , cos π 2 - α + 2 π z = sin α t g π 2 - α + 2 π z = c t g α , c t g π 2 - α + 2 π z = t g α sin π + α + 2 π z = - sin α , cos π + α + 2 π z = - cos α t g π + α + 2 π z = t g α , c t g π + α + 2 π z = c t g α sin π - α + 2 π z = sin α , cos π - α + 2 π z = - cos α t g π - α + 2 π z = - t g α , c t g π - α + 2 π z = - c t g α sin 3 π 2 + α + 2 π z = - cos α , cos 3 π 2 + α + 2 π z = sin α t g 3 π 2 + α + 2 π z = - c t g α , c t g 3 π 2 + α + 2 π z = - t g α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α t g 3 π 2 - α + 2 π z = c t g α , c t g 3 π 2 - α + 2 π z = t g α
In this case, the formulas are written in radians. However, you can also write them using degrees. It is enough just to convert radians to degrees, replacing π by 180 degrees.
Examples of using reduction formulas
We will show how to use reduction formulas and how these formulas are used to solve practical examples.
The angle under the sign of the trigonometric function can be represented not in one, but in many ways. For example, the argument of a trigonometric function can be represented in the form ± α + 2 π z, π 2 ± α + 2 π z, π ± α + 2 π z, 3 π 2 ± α + 2 π z. Let's demonstrate this.
Let's take the angle α = 16 π 3. This angle can be written like this:
α = 16 π 3 = π + π 3 + 2 π 2 α = 16 π 3 = - 2 π 3 + 2 π 3 α = 16 π 3 = 3 π 2 - π 6 + 2 π
Depending on the representation of the angle, the appropriate reduction formula is used.
Let's take the same angle α = 16 π 3 and calculate its tangent
Example 1: Using reduction formulas
α = 16 π 3 , t g α = ?
Let us represent the angle α = 16 π 3 as α = π + π 3 + 2 π 2
This representation of the angle will correspond to the reduction formula
t g (π + α + 2 π z) = t g α
t g 16 π 3 = t g π + π 3 + 2 π 2 = t g π 3
Using the table, we indicate the value of the tangent
Now we use another representation of the angle α = 16 π 3.
Example 2: Using reduction formulas
α = 16 π 3 , t g α = ? α = - 2 π 3 + 2 π 3 t g 16 π 3 = t g - 2 π 3 + 2 π 3 = - t g 2 π 3 = - (- 3) = 3
Finally, for the third representation of the angle we write
Example 3. Using reduction formulas
α = 16 π 3 = 3 π 2 - π 6 + 2 π t g 3 π 2 - α + 2 π z = c t g α t g α = t g (3 π 2 - π 6 + 2 π) = c t g π 6 = 3
Now let's give an example of using more complex reduction formulas
Example 4: Using reduction formulas
Let's imagine sin 197° through the sine and cosine of an acute angle.
In order to be able to apply reduction formulas, you need to represent the angle α = 197 ° in one of the forms
± α + 360 ° z, 90 ° ± α + 360 ° z, 180 ° ± α + 360 ° z, 270 ° ± α + 360 ° z. According to the conditions of the problem, the angle must be acute. Accordingly, we have two ways to represent it:
197° = 180° + 17° 197° = 270° - 73°
We get
sin 197° = sin (180° + 17°) sin 197° = sin (270° - 73°)
Now let's look at the reduction formulas for sines and choose the appropriate ones
sin (π + α + 2 πz) = - sinα sin (3 π 2 - α + 2 πz) = - cosα sin 197 ° = sin (180 ° + 17 ° + 360 ° z) = - sin 17 ° sin 197 ° = sin (270 ° - 73 ° + 360 ° z) = - cos 73 °
Mnemonic rule
There are many reduction formulas, and, fortunately, there is no need to memorize them. There are regularities by which reduction formulas can be derived for different angles and trigonometric functions. These patterns are called mnemonic rules. Mnemonics is the art of memorization. The mnemonic rule consists of three parts, or contains three stages.
Mnemonic rule
1. The argument of the original function is represented in one of the following forms:
± α + 2 πz π 2 ± α + 2 πz π ± α + 2 πz 3 π 2 ± α + 2 πz
Angle α must lie between 0 and 90 degrees.
2. The sign of the original trigonometric function is determined. The function written on the right side of the formula will have the same sign.
3. For angles ± α + 2 πz and π ± α + 2 πz the name of the original function remains unchanged, and for angles π 2 ± α + 2 πz and 3 π 2 ± α + 2 πz, respectively, it changes to “cofunction”. Sine - cosine. Tangent - cotangent.
To use the mnemonic guide for reduction formulas, you need to be able to determine the signs of trigonometric functions based on the quarters of the unit circle. Let's look at examples of using the mnemonic rule.
Example 1: Using a mnemonic rule
Let's write down the reduction formulas for cos π 2 - α + 2 πz and t g π - α + 2 πz. α is the log of the first quarter.
1. Since by condition α is the log of the first quarter, we skip the first point of the rule.
2. Define the signs cos functionsπ 2 - α + 2 πz and t g π - α + 2 πz. The angle π 2 - α + 2 πz is also the angle of the first quarter, and the angle π - α + 2 πz is in the second quarter. In the first quarter, the cosine function is positive, and the tangent in the second quarter has a minus sign. Let's write down what the required formulas will look like at this stage.
cos π 2 - α + 2 πz = + t g π - α + 2 πz = -
3. According to the third point, for the angle π 2 - α + 2 π the name of the function changes to Confucius, and for the angle π - α + 2 πz remains the same. Let's write down:
cos π 2 - α + 2 πz = + sin α t g π - α + 2 πz = - t g α
Now let’s look at the formulas given above and make sure that the mnemonic rule works.
Let's look at an example with a specific angle α = 777°. Let us reduce sine alpha to the trigonometric function of an acute angle.
Example 2: Using a mnemonic rule
1. Imagine the angle α = 777 ° in required form
777° = 57° + 360° 2 777° = 90° - 33° + 360° 2
2. The original angle is the angle of the first quarter. This means that the sine of the angle has a positive sign. As a result we have:
3. sin 777° = sin (57° + 360° 2) = sin 57° sin 777° = sin (90° - 33° + 360° 2) = cos 33°
Now let's look at an example that shows how important it is to correctly determine the sign of the trigonometric function and correctly represent the angle when using the mnemonic rule. Let's repeat it again.
Important!
Angle α must be acute!
Let's calculate the tangent of the angle 5 π 3. From the table of values of the main trigonometric functions, you can immediately take the value t g 5 π 3 = - 3, but we will apply the mnemonic rule.
Example 3: Using a mnemonic rule
Let's imagine the angle α = 5 π 3 in the required form and use the rule
t g 5 π 3 = t g 3 π 2 + π 6 = - c t g π 6 = - 3 t g 5 π 3 = t g 2 π - π 3 = - t g π 3 = - 3
If we represent the alpha angle in the form 5 π 3 = π + 2 π 3, then the result of applying the mnemonic rule will be incorrect.
t g 5 π 3 = t g π + 2 π 3 = - t g 2 π 3 = - (- 3) = 3
The incorrect result is due to the fact that the angle 2 π 3 is not acute.
The proof of the reduction formulas is based on the properties of periodicity and symmetry of trigonometric functions, as well as on the property of shift by angles π 2 and 3 π 2. The proof of the validity of all reduction formulas can be carried out without taking into account the term 2 πz, since it denotes a change in the angle by an integer number of full revolutions and precisely reflects the property of periodicity.
The first 16 formulas follow directly from the properties of the basic trigonometric functions: sine, cosine, tangent and cotangent.
Here is a proof of the reduction formulas for sines and cosines
sin π 2 + α = cos α and cos π 2 + α = - sin α
Let's look at a unit circle, the starting point of which, after a rotation through an angle α, goes to the point A 1 x, y, and after a rotation through an angle π 2 + α - to a point A 2. From both points we draw perpendiculars to the abscissa axis.
Two right triangles O A 1 H 1 and O A 2 H 2 are equal in hypotenuse and adjacent angles. From the location of points on the circle and the equality of triangles, we can conclude that point A 2 has coordinates A 2 - y, x. Using the definitions of sine and cosine, we write:
sin α = y, cos α = x, sin π 2 + α = x, cos π 2 + α = y
sin π 2 + α = cos α, cos π 2 + α = - sin α
Taking into account the basic identities of trigonometry and what has just been proven, we can write
t g π 2 + α = sin π 2 + α cos π 2 + α = cos α - sin α = - c t g α c t g π 2 + α = cos π 2 + α sin π 2 + α = - sin α cos α = - t g α
To prove reduction formulas with argument π 2 - α, it must be presented in the form π 2 + (- α). For example:
cos π 2 - α = cos π 2 + (- α) = - sin (- α) = sin α
The proof uses the properties of trigonometric functions with arguments of opposite signs.
All other reduction formulas can be proven based on those written above.
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Lesson topic
- Changes in sine, cosine and tangent as the angle increases.
Lesson Objectives
- Get acquainted with new definitions and remember some already studied.
- Get acquainted with the pattern of changes in the values of sine, cosine and tangent as the angle increases.
- Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson Objectives
- Test students' knowledge.
Lesson Plan
- Repetition of previously studied material.
- Repetition tasks.
- Changes in sine, cosine and tangent as the angle increases.
- Practical use.
Repetition of previously studied material
Let's start from the very beginning and remember what will be useful to refresh your memory. What are sine, cosine and tangent and what branch of geometry do these concepts belong to?
Trigonometry- it's so complicated Greek word: trigonon - triangle, metro - to measure. Therefore, in Greek this means: measured by triangles.
Subjects > Mathematics > Mathematics 8th gradeHow to remember formulas for reducing trigonometric functions? It's easy if you use an association. This association was not invented by me. As already stated, good association It should “catch”, that is, evoke vivid emotions. I cannot call the emotions caused by this association positive. But it gives a result - it allows you to remember reduction formulas, which means it has the right to exist. After all, if you don't like it, you don't have to use it, right?
The reduction formulas have the form: sin(πn/2±α), cos(πn/2±α), tg(πn/2±α), ctg(πn/2±α). Remember that +α gives counterclockwise movement, - α gives clockwise movement.
To work with reduction formulas, you need two points:
1) put the sign that the initial function has (in textbooks they write: reducible. But in order not to get confused, it is better to call it initial), if we consider α to be the angle of the first quarter, that is, small.
2) Horizontal diameter - π±α, 2π±α, 3π±α... - in general, when there is no fraction, the name of the function does not change. Vertical π/2±α, 3π/2±α, 5π/2±α... - when there is a fraction, the name of the function changes: sine - to cosine, cosine - to sine, tangent - to cotangent and cotangent - to tangent. 
Now, actually, the association:
vertical diameter (there is a fraction) -
standing drunk. What will happen to him early?
or is it too late? That's right, it will fall.
The function name will change.
If the diameter is horizontal, the drunk is already lying down. He's probably sleeping. Nothing will happen to him; he has already assumed a horizontal position. Accordingly, the name of the function does not change.
That is, sin(π/2±α), sin(3π/2±α), sin(5π/2±α), etc. give ±cosα,
and sin(π±α), sin(2π±α), sin(3π±α), … - ±sinα.
We already know how.
How it works? Let's look at examples.
1) cos(π/2+α)=?
We become π/2. Since +α means we go forward, counterclockwise. We find ourselves in the second quarter, where the cosine has a “-“ sign. The name of the function changes (“a drunk person is standing”, which means he will fall). So,
cos(π/2+α)=-sin α.
Let's get to 2π. Since -α - we go backwards, that is, clockwise. We find ourselves in the IV quarter, where the tangent has a “-“ sign. The name of the function does not change (the diameter is horizontal, “the drunk is already lying down”). Thus, tan(2π-α)=- tanα.
3) ctg²(3π/2-α)=?
Examples in which a function is raised to an even power are even simpler to solve. The even degree “-” removes it, that is, you just need to find out whether the name of the function changes or remains. The diameter is vertical (there is a fraction, “standing drunk”, it will fall), the name of the function changes. We get: ctg²(3π/2-α)= tan²α.
Definition. Reduction formulas are formulas that allow you to move from trigonometric functions of the form to functions of argument. With their help, the sine, cosine, tangent and cotangent of an arbitrary angle can be reduced to the sine, cosine, tangent and cotangent of an angle from the interval from 0 to 90 degrees (from 0 to radians). Thus, reduction formulas allow us to move on to working with angles within 90 degrees, which is undoubtedly very convenient.
Reduction formulas:
There are two rules for using reduction formulas.
1. If the angle can be represented as (π/2 ±a) or (3*π/2 ±a), then function name changes sin to cos, cos to sin, tg to ctg, ctg to tg. If the angle can be represented in the form (π ±a) or (2*π ±a), then The function name remains unchanged.
Look at the picture below, it shows schematically when to change the sign and when not
2. Sign of the reduced function remains the same. If the original function had a plus sign, then the reduced function also has a plus sign. If the original function had a minus sign, then the reduced function also has a minus sign.
The figure below shows the signs of the basic trigonometric functions depending on the quarter.
Example:
Calculate
Let's use the reduction formulas:
Sin(150˚) is in the second quarter; from the figure we see that the sin sign in this quarter is equal to “+”. This means that the given function will also have a “+” sign. We applied the second rule.
Now 150˚ = 90˚ +60˚. 90˚ is π/2. That is, we are dealing with the case π/2+60, therefore, according to the first rule, we change the function from sin to cos. As a result, we get Sin(150˚) = cos(60˚) = ½.
Centered at a point A.
α
- angle expressed in radians.
Definition
Sine (sin α)- This trigonometric function, depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio length of the opposite side |BC| to the length of the hypotenuse |AC|.
Cosine (cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the hypotenuse |AC|.
Accepted notations
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Graph of the sine function, y = sin x
Graph of the cosine function, y = cos x
Properties of sine and cosine
Periodicity
Functions y = sin x and y = cos x periodic with period 2π.
Parity
The sine function is odd. The cosine function is even.
Domain of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see proof of continuity). Their main properties are presented in the table (n - integer).
| y = sin x | y = cos x | |
| Scope and continuity | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | -1 ≤ y ≤ 1 | -1 ≤ y ≤ 1 |
| Increasing | ||
| Descending | ||
| Maxima, y = 1 | ||
| Minima, y = - 1 | ||
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y = 0 | y = 1 |
Basic formulas
Sum of squares of sine and cosine
Formulas for sine and cosine from sum and difference
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Formulas for the product of sines and cosines
Sum and difference formulas
Expressing sine through cosine
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Expressing cosine through sine
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Expression through tangent
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When , we have:
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At :
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Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument. 
Expressions through complex variables
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Euler's formula
Expressions through hyperbolic functions
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Derivatives
; . Deriving formulas > > >
Derivatives of nth order:
{ -∞ <
x < +∞ }
Secant, cosecant
Inverse functions
The inverse functions of sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
Arccosine, arccos
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.