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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 a good quality photo to our email address, and we will make the necessary duplicate.
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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.
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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.
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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.
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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
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.
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Trigonometry is one of the branches of mathematics, the study of which focuses on angles and the relationships between them. The foundations of science are laid in school years, when definitions of angle functions are introduced. In the future, the resulting base is used in the development of astronomy, instrumentation, architecture and other areas of knowledge. Like any exact science, trigonometry is not complete without formulas. Practical applications have found expressions for the definition of a double argument. For example, by resorting to the corresponding equation, you can easily find out the double angle of the sine.
Trigonometric expression for calculation
The expression is simply written and remembered: the sine of a double angle is calculated as the double product of the sine and cosine of a single argument.
This formula is derived from the expression for the sine of the sum of the angles ( Q 1 + Q 2 ) :
sin( Q 1 + Q 2) = sin Q 1* cos Q 1+ sin Q 2*cos Q 2 .
Assuming that the given angles are equal to each other, the formula is written in the usual form.
You can use an expression for any value of the function argument. Calculating the double angle of the sine from it is quite simple, the examples below will help to verify this.
Usage example
Here are some illustrations of the application of the resulting formula. Let it be required to calculate the value of the trigonometric function of the sine of an angle equal to 60 degrees. The corresponding single angle would be 30 degrees. Since the values of the sine and cosine of the 30 degree angle are known, the double angle of the sine will be sin 60 = 2 * sin 30 * cos 30.
The formula is used not only for calculating "manually", you can also find values using it using mathematical packages or MS Excel tables.
Despite the simplicity of the trigonometric identity, it causes difficulties for school graduates. This is exactly what the developers of the USE tasks are counting on, offering tests to check the basic formulas. Conclusion - the formula to calculate the double angle of the sine, you need to know by heart!
Centered at a point A.
α
is an angle expressed in radians.
Definition
Sinus 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 opposite leg |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 designations
;
;
.
;
;
.
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 a period 2 pi.
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 on their domain of definition, that is, for all x (see the 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 |
| Ascending | ||
| Descending | ||
| Maximums, y= 1 | ||
| Minima, y = - 1 | ||
| Zeros, y= 0 | ||
| Points of intersection with the y-axis, x = 0 | y= 0 | y= 1 |
Basic Formulas
Sum of squared sine and cosine
Sine and cosine formulas for sum and difference
;
;
Formulas for the product of sines and cosines
Sum and difference formulas
Expression of sine through cosine
;
;
;
.
Expression of cosine through sine
;
;
;
.
Expression in terms of tangent
; .
For , we have:
;
.
At :
;
.
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for some values of the argument. 
Expressions through complex variables
;
Euler formula
Expressions in terms of hyperbolic functions
;
;
Derivatives
; . Derivation of formulas > > >
Derivatives of the nth order:
{ -∞ <
x < +∞ }
Secant, cosecant
Inverse functions
The inverse functions to 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 Students of Higher Educational Institutions, Lan, 2009.
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 a good quality photo to our email address, 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
Valentine:
You saved our son from being fired! The fact is that after dropping out of school, the son went into the army. And when he returned, he did not want to recover. Worked without a degree. But recently they began to fire everyone who does not have a “crust. Therefore, we decided to contact you and did not regret it! Now he works calmly and is not afraid of anything! Thank you!
