Most frequently asked questions
Is it possible to make a seal on a document according to the provided sample? Answer Yes, it's possible. Send a scanned copy or photo to our email address good quality and we will make the necessary duplicate.
What types of payment do you accept?
Answer You can pay for the document at the time of receipt by the courier, after you check the correctness of filling and the quality of the diploma. This can also be done at the office of postal companies offering cash on delivery services.
All terms of delivery and payment of documents are described in the section "Payment and Delivery". We are also ready to listen to your suggestions on the terms of delivery and payment for the document.
Can I be sure that after placing an order you will not disappear with my money? Answer We have quite a long experience in the field of diploma production. We have several sites that are constantly updated. Our specialists work in different parts of the country, producing over 10 documents a day. Over the years, our documents have helped many people solve employment problems or move to higher paying jobs. We have earned trust and recognition among customers, so there is absolutely no reason for us to do this. Moreover, it is simply impossible to do it physically: you pay for your order at the time of receiving it in your hands, there is no prepayment.
Can I order a diploma from any university? Answer In general, yes. We have been working in this area for almost 12 years. During this time, an almost complete database of documents issued by almost all universities in the country and for different years of issue has been formed. All you need is to choose a university, specialty, document, and fill out an order form.
What should I do if I find typos and errors in a document?
Answer When receiving a document from our courier or postal company, we recommend that you carefully check all the details. If a typo, error or inaccuracy is found, you have the right not to take the diploma, and you must indicate the shortcomings found personally to the courier or in writing by sending an e-mail.
As soon as possible, we will correct the document and resend it to the specified address. Of course, the shipping will be paid by our company.
To avoid such misunderstandings, before filling out the original form, we send a layout of the future document to the customer's mail for verification and approval of the final version. Before sending the document by courier or mail, we also take an additional photo and video (including in ultraviolet light) so that you have a visual idea of what you will get in the end.
What do you need to do to order a diploma from your company?
Answer To order a document (certificate, diploma, academic certificate, etc.), you must fill out an online order form on our website or provide your e-mail so that we send you a questionnaire form, which you need to fill out and send back to us.
If you do not know what to indicate in any field of the order form/questionnaire, leave them blank. Therefore, we will clarify all the missing information over the phone.
Latest reviews
Alexei:
I needed to get a diploma to get a job as a manager. And most importantly, I have both experience and skills, but without a document I can’t, I’ll get a job anywhere. Once on your site, I still decided to buy a diploma. The diploma was completed in 2 days! Now I have a job that I never dreamed of before!! Thank you!
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 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 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
main destination 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.
Copyright by clever students
All rights reserved.
Protected by copyright law. No part of www.site, including internal materials and external design, may be reproduced in any form or used without the prior written permission of the copyright holder.
I will not convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and how cheat sheets are useful. And here - information on how not to learn, but to remember some trigonometric formulas. So - trigonometry without a cheat sheet! We use associations for memorization.
1. Addition formulas:
cosines always "go in pairs": cosine-cosine, sine-sine.
And one more thing: cosines are “inadequate”. They “everything is wrong”, so they change the signs: “-” to “+”, and vice versa.
Sinuses - "mix": sine-cosine, cosine-sine.
2. Sum and difference formulas:
cosines always "go in pairs". Having added two cosines - "buns", we get a pair of cosines - "koloboks". And subtracting, we definitely won’t get koloboks. We get a couple of sines. Still with a minus ahead.
Sinuses - "mix" :
3. Formulas for converting a product into a sum and a difference.
When do we get a pair of cosines? When adding the cosines. That's why
When do we get a pair of sines? When subtracting cosines. From here:
"Mixing" is obtained both by adding and subtracting sines. Which is more fun: adding or subtracting? That's right, fold. And for the formula take addition:
In the first and third formulas in brackets - the amount. From the rearrangement of the places of the terms, the sum does not change. The order is important only for the second formula. But, in order not to get confused, for ease of remembering, in all three formulas in the first brackets we take the difference
and secondly, the sum
Crib sheets in your pocket give peace of mind: if you forget the formula, you can write it off. And they give confidence: if you fail to use the cheat sheet, the formulas can be easily remembered.
The formulas for the sum and difference of sines and cosines for two angles α and β allow you to go from the sum of the indicated angles to the product of the angles α + β 2 and α - β 2 . We note right away that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for sines and cosines of the sum and difference. Below we list these formulas, give their derivation and show examples of application for specific problems.
Yandex.RTB R-A-339285-1
Formulas for the sum and difference of sines and cosines
Let's write down how the sum and difference formulas for sines and cosines look like
Sum and difference formulas for sines
sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2
Sum and difference formulas for cosines
cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 cos α - β 2, cos α - cos β = 2 sin α + β 2 β -α 2
These formulas are valid for any angles α and β. The angles α + β 2 and α - β 2 are called, respectively, the half-sum and half-difference of the angles alpha and beta. We give a formulation for each formula.
Definitions of sum and difference formulas for sines and cosines
The sum of the sines of two angles is equal to twice the product of the sine of the half-sum of these angles and the cosine of the half-difference.
Difference of sines of two angles is equal to twice the product of the sine of the half-difference of these angles and the cosine of the half-sum.
The sum of the cosines of two angles is equal to twice the product of the cosine of the half-sum and the cosine of the half-difference of these angles.
Difference of cosines of two angles is equal to twice the product of the sine of the half-sum and the cosine of the half-difference of these angles, taken with a negative sign.
Derivation of formulas for the sum and difference of sines and cosines
To derive formulas for the sum and difference of the sine and cosine of two angles, addition formulas are used. We present them below
sin (α + β) = sin α cos β + cos α sin β sin (α - β) = sin α cos β - cos α sin β cos (α + β) = cos α cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β
We also represent the angles themselves as the sum of half-sums and half-differences.
α \u003d α + β 2 + α - β 2 \u003d α 2 + β 2 + α 2 - β 2 β \u003d α + β 2 - α - β 2 \u003d α 2 + β 2 - α 2 + β 2
We proceed directly to the derivation of the sum and difference formulas for sin and cos.
Derivation of the formula for the sum of sines
In the sum sin α + sin β, we replace α and β with the expressions for these angles given above. Get
sin α + sin β = sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2
Now we apply the addition formula to the first expression, and the sine formula of the angle differences to the second one (see the formulas above)
sin α + β 2 + α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2
sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α + β 2 cos α - β 2
The steps for deriving the rest of the formulas are similar.
Derivation of the formula for the difference of sines
sin α - sin β = sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 - sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α - β 2 cos α + β 2
Derivation of the formula for the sum of cosines
cos α + cos β = cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 + cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = 2 cos α + β 2 cos α - β 2
Derivation of the cosine difference formula
cos α - cos β = cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 - cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = - 2 sin α + β 2 sin α - β 2
Examples of solving practical problems
To begin with, we will check one of the formulas by substituting specific angle values into it. Let α = π 2 , β = π 6 . Let's calculate the value of the sum of the sines of these angles. First, we use the table of basic values of trigonometric functions, and then we apply the formula for the sum of sines.
Example 1. Checking the formula for the sum of the sines of two angles
α \u003d π 2, β \u003d π 6 sin π 2 + sin π 6 \u003d 1 + 1 2 \u003d 3 2 sin π 2 + sin π 6 \u003d 2 sin π 2 + π 6 2 cos π 2 - π 6 2 \u003d 2 sin π 3 cos π 6 \u003d 2 3 2 3 2 \u003d 3 2
Let us now consider the case when the values of the angles differ from the basic values presented in the table. Let α = 165°, β = 75°. Let us calculate the value of the difference between the sines of these angles.
Example 2. Applying the sine difference formula
α = 165 ° , β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - sin 75 ° 2 cos 165 ° + sin 75 ° 2 = = 2 sin 45 ° cos 120 ° = 2 2 2 - 1 2 = 2 2
Using the formulas for the sum and difference of sines and cosines, you can go from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for the transition from sum to product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and in converting trigonometric expressions.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were developed by astronomers to create an accurate calendar and orientate by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of the sides and angle of a flat triangle.
Trigonometry is a branch of mathematics dealing with the properties of trigonometric functions and the relationship between the sides and angles of triangles.
During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi introduced such functions as tangent and cotangent, compiled the first tables of values for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.
Basic quantities of trigonometry
The basic trigonometric functions of a numerical argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.
The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants, equal in all directions,” since the proof is given on the example of an isosceles right triangle.
Sine, cosine and other dependencies establish a relationship between acute angles and sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:
As you can see, tg and ctg are inverse functions. If we represent leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, then we get the following formulas for tangent and cotangent:
trigonometric circle
Graphically, the ratio of the mentioned quantities can be represented as follows:
The circle, in this case, represents all possible values of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a “+” sign if α belongs to the I and II quarters of the circle, that is, it is in the range from 0 ° to 180 °. With α from 180° to 360° (III and IV quarters), sin α can only be a negative value.
Let's try to build trigonometric tables for specific angles and find out the meaning of the quantities.
The values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.
These angles were not chosen by chance. The designation π in the tables is for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal relationship; when calculating in radians, the actual length of the radius in cm does not matter.
The angles in the tables for trigonometric functions correspond to radian values:
So, it is not difficult to guess that 2π is a full circle or 360°.
Properties of trigonometric functions: sine and cosine
In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.
Consider a comparative table of properties for a sine wave and a cosine wave:
| sinusoid | cosine wave |
|---|---|
| y = sin x | y = cos x |
| ODZ [-1; one] | ODZ [-1; one] |
| sin x = 0, for x = πk, where k ϵ Z | cos x = 0, for x = π/2 + πk, where k ϵ Z |
| sin x = 1, for x = π/2 + 2πk, where k ϵ Z | cos x = 1, for x = 2πk, where k ϵ Z |
| sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z | cos x = - 1, for x = π + 2πk, where k ϵ Z |
| sin (-x) = - sin x, i.e. odd function | cos (-x) = cos x, i.e. the function is even |
| the function is periodic, the smallest period is 2π | |
| sin x › 0, with x belonging to quarters I and II or from 0° to 180° (2πk, π + 2πk) | cos x › 0, with x belonging to quarters I and IV or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk) |
| sin x ‹ 0, with x belonging to quarters III and IV or from 180° to 360° (π + 2πk, 2π + 2πk) | cos x ‹ 0, with x belonging to quarters II and III or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk) |
| increases on the interval [- π/2 + 2πk, π/2 + 2πk] | increases on the interval [-π + 2πk, 2πk] |
| decreases on the intervals [ π/2 + 2πk, 3π/2 + 2πk] | decreases in intervals |
| derivative (sin x)' = cos x | derivative (cos x)’ = - sin x |
Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs are the same, the function is even; otherwise, it is odd.
The introduction of radians and the enumeration of the main properties of the sinusoid and cosine wave allow us to bring the following pattern:
It is very easy to verify the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. Checking can be done by looking at tables or by tracing function curves for given values.
Properties of tangentoid and cotangentoid
The graphs of the tangent and cotangent functions differ significantly from the sinusoid and cosine wave. The values tg and ctg are inverse to each other.
- Y = tgx.
- The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
- The smallest positive period of the tangentoid is π.
- Tg (- x) \u003d - tg x, i.e., the function is odd.
- Tg x = 0, for x = πk.
- The function is increasing.
- Tg x › 0, for x ϵ (πk, π/2 + πk).
- Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
- Derivative (tg x)' = 1/cos 2 x .
Consider the graphical representation of the cotangentoid below in the text.
The main properties of the cotangentoid:
- Y = ctgx.
- Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
- The cotangentoid tends to the values of y at x = πk, but never reaches them.
- The smallest positive period of the cotangentoid is π.
- Ctg (- x) \u003d - ctg x, i.e., the function is odd.
- Ctg x = 0, for x = π/2 + πk.
- The function is decreasing.
- Ctg x › 0, for x ϵ (πk, π/2 + πk).
- Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
- Derivative (ctg x)' = - 1/sin 2 x Fix
