Trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known.

tg \alpha = \frac(\sin \alpha)(\cos \alpha), \enspace ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

tg \alpha \cdot ctg \alpha = 1

This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa.

When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in reverse order.

Finding tangent and cotangent through sine and cosine

tg \alpha = \frac(\sin \alpha)(\cos \alpha),\enspace

These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look, then by definition, the ordinate of y is the sine, and the abscissa of x is the cosine. Then the tangent will be equal to the ratio \frac(y)(x)=\frac(\sin \alpha)(\cos \alpha), and the ratio \frac(x)(y)=\frac(\cos \alpha)(\sin \alpha)- will be a cotangent.

We add that only for such angles \alpha for which the trigonometric functions included in them make sense, the identities will take place, ctg \alpha=\frac(\cos \alpha)(\sin \alpha).

For example: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for \alpha angles that are different from \frac(\pi)(2)+\pi z, a ctg \alpha=\frac(\cos \alpha)(\sin \alpha)- for an angle \alpha other than \pi z , z is an integer.

Relationship between tangent and cotangent

tg \alpha \cdot ctg \alpha=1

This identity is valid only for angles \alpha that are different from \frac(\pi)(2) z. Otherwise, either cotangent or tangent will not be determined.

Based on the points above, we get that tg \alpha = \frac(y)(x), a ctg\alpha=\frac(x)(y). Hence it follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of one angle at which they make sense are mutually reciprocal numbers.

Relationships between tangent and cosine, cotangent and sine

tg^(2) \alpha + 1=\frac(1)(\cos^(2) \alpha)- the sum of the square of the tangent of the angle \alpha and 1 is equal to the inverse square of the cosine of this angle. This identity is valid for all \alpha other than \frac(\pi)(2)+ \pi z.

1+ctg^(2) \alpha=\frac(1)(\sin^(2)\alpha)- the sum of 1 and the square of the cotangent of the angle \alpha , equals the inverse square of the sine of the given angle. This identity is valid for any \alpha other than \pi z .

Examples with solutions to problems using trigonometric identities

Example 1

Find \sin \alpha and tg \alpha if \cos \alpha=-\frac12 and \frac(\pi)(2)< \alpha < \pi ;

Show Solution

Decision

The functions \sin \alpha and \cos \alpha are linked by the formula \sin^(2)\alpha + \cos^(2) \alpha = 1. Substituting into this formula \cos \alpha = -\frac12, we get:

\sin^(2)\alpha + \left (-\frac12 \right)^2 = 1

This equation has 2 solutions:

\sin \alpha = \pm \sqrt(1-\frac14) = \pm \frac(\sqrt 3)(2)

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the sine is positive, so \sin \alpha = \frac(\sqrt 3)(2).

To find tg \alpha , we use the formula tg \alpha = \frac(\sin \alpha)(\cos \alpha)

tg \alpha = \frac(\sqrt 3)(2) : \frac12 = \sqrt 3

Example 2

Find \cos \alpha and ctg \alpha if and \frac(\pi)(2)< \alpha < \pi .

Show Solution

Decision

Substituting into the formula \sin^(2)\alpha + \cos^(2) \alpha = 1 conditional number \sin \alpha=\frac(\sqrt3)(2), we get \left (\frac(\sqrt3)(2)\right)^(2) + \cos^(2) \alpha = 1. This equation has two solutions \cos \alpha = \pm \sqrt(1-\frac34)=\pm\sqrt\frac14.

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the cosine is negative, so \cos \alpha = -\sqrt\frac14=-\frac12.

In order to find ctg \alpha , we use the formula ctg \alpha = \frac(\cos \alpha)(\sin \alpha). We know the corresponding values.

ctg \alpha = -\frac12: \frac(\sqrt3)(2) = -\frac(1)(\sqrt 3).

In this article, we will take a comprehensive look at . Basic trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.

We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations.

Page navigation.

Relationship between sine and cosine of one angle

Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind . The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both of its parts by and respectively, and the equalities and follow from the definitions of sine, cosine, tangent, and cotangent. We will discuss this in more detail in the following paragraphs.

That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.

Before proving the basic trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.

The basic trigonometric identity is very often used in transformation of trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle.

Tangent and cotangent through sine and cosine

Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the tangent is the ratio of the ordinate to the abscissa, that is, , and the cotangent is the ratio of the abscissa to the ordinate, that is, .

Due to this obviousness of the identities and often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.

To conclude this section, it should be noted that the identities and hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be zero, and we did not define division by zero), and the formula - for all , different from , where z is any .

Relationship between tangent and cotangent

An even more obvious trigonometric identity than the two previous ones is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.

Proof of the formula very simple. By definition and from where . The proof could have been carried out in a slightly different way. Since and , then .

So, the tangent and cotangent of one angle, at which they make sense, is.

In this article, we will talk about universal trigonometric substitution. It involves the expression of the sine, cosine, tangent and cotangent of any angle through the tangent of a half angle. Moreover, such a replacement is carried out rationally, that is, without roots.

First, we write formulas expressing the sine, cosine, tangent and cotangent in terms of the tangent of a half angle. Next, we show the derivation of these formulas. And in conclusion, let's look at several examples of using the universal trigonometric substitution.

Page navigation.

Sine, cosine, tangent and cotangent through the tangent of a half angle

First, let's write down four formulas expressing the sine, cosine, tangent and cotangent of an angle in terms of the tangent of a half angle.

These formulas are valid for all angles at which the tangents and cotangents included in them are defined:

Derivation of formulas

Let us analyze the derivation of formulas expressing the sine, cosine, tangent and cotangent of an angle through the tangent of a half angle. Let's start with the formulas for sine and cosine.

We represent the sine and cosine using the double angle formulas as and respectively. Now expressions and write as fractions with denominator 1 as and . Further, on the basis of the main trigonometric identity, we replace the units in the denominator with the sum of the squares of the sine and cosine, after which we obtain and . Finally, we divide the numerator and denominator of the resulting fractions by (its value is different from zero, provided ). As a result, the whole chain of actions looks like this:


and

This completes the derivation of formulas expressing the sine and cosine through the tangent of a half angle.

It remains to derive the formulas for the tangent and cotangent. Now, taking into account the formulas obtained above, and the formulas and , we immediately obtain formulas expressing the tangent and cotangent through the tangent of a half angle:

So, we have derived all the formulas for the universal trigonometric substitution.

Examples of using the universal trigonometric substitution

First, let's consider an example of using universal trigonometric substitution when converting expressions.

Example.

Give an expression to an expression containing only one trigonometric function.

Decision.

Answer:

.

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.

Instruction

Use your knowledge of planimetry to express sinus through co sinus. By definition, sinus ohm of an angle in a right triangle of length opposite to, and to sinus om - the adjacent leg to the hypotenuse. Even knowledge of the Pythagorean theorem will allow you to quickly find the desired transformation in some cases.

express sinus through co sinus, using the simplest trigonometric identity, according to which the sum of the squares of these quantities gives unity. Please note that you can correctly complete the task only if you know that the desired angle is in the quarter, otherwise you will get two possible results - with a positive and a sign.

cos?=(b?+c?-a?)/(2*b*c)

There is a triangle with sides a, b, c equal to 3, 4, 5 mm, respectively.

To find cosine the angle enclosed between the large sides.

Let us denote the angle opposite to the side a through?, then, according to the formula derived above, we have:

cos?=(b?+c?-a?)/(2*b*c)=(4?+5?-3?)/(2*4*5)=(16+25-9)/40 =32/40=0.8

Answer: 0.8.

If the triangle is a right triangle, then to find cosine and it is enough to know the lengths of any two sides of the angle ( cosine right angle is 0).

Let there be a right triangle with sides a, b, c, where c is the hypotenuse.

Consider all options:

Find cos? if the lengths of the sides a and b (of a triangle) are known

Let's use additionally the Pythagorean theorem:

cos?=(b?+c?-a?)/(2*b*c)=(b?+b?+a?-a?)/(2*b*v(b?+a?)) =(2*b?)/(2*b*v(b?+a?))=b/v(b?+a?)

In order for the correctness of the resulting formula, we substitute into it from example 1, i.e.

Having done elementary calculations, we get:

Similarly, there is cosine in a rectangular triangle in other cases:

Known a and c (hypotenuse and opposite leg), find cos?

cos?=(b?+c?-a?)/(2*b*c)=(c?-a?+c?-a?)/(2*c*v(c?-a?)) =(2*s?-2*a?)/(2*s*v(s?-a?))=v(s?-a?)/s.

Substituting the values ​​a=3 and c=5 from the example, we get:

b and c are known (the hypotenuse and the adjacent leg).

Find cos?

Having performed similar transformations (shown in examples 2 and 3), we obtain that in this case cosine in triangle calculated using a very simple formula:

The simplicity of the derived formula is explained in an elementary way: in fact, adjacent to the corner? the leg is a projection of the hypotenuse, its length is equal to the length of the hypotenuse multiplied by cos?.

Substituting the values ​​b=4 and c=5 from the first example, we get:

So all our formulas are correct.

In order to obtain a formula relating sinus and co sinus angle, it is necessary to give or recall some definitions. So, sinus angle is the ratio (quotient of division) of the opposite leg of a right triangle to the hypotenuse. Co. sinus angle is the ratio of the adjacent leg to the hypotenuse.

Instruction

Helpful advice

The value of the sine and cosine of any angle cannot be greater than 1.

Sinus and cosine- these are direct trigonometric functions for which there are several definitions - through a circle in a Cartesian coordinate system, through solutions of a differential equation, through acute angles in a right triangle. Each of these definitions allows you to deduce the relationship between these two functions. The following is perhaps the simplest way to express cosine through the sine - through their definitions for acute angles of a right triangle.

Instruction

Express the sine of an acute angle of a right triangle in terms of the lengths of the sides of this figure. According to the definition, the sine of the angle (α) must be the ratio of the length of the side (a) opposite it - the leg - to the length of the side (c) opposite the right angle - the hypotenuse: sin (α) = a / c.

Find a similar formula for cosine but the same angle. By definition, this value should be expressed as the ratio of the length of the side (b) adjacent to this corner (the second leg) to the length of the side (c) lying opposite the right angle: cos (a) \u003d a / c.

Rewrite the equation following from the Pythagorean theorem in such a way that it uses the relations between the legs and the hypotenuse derived in the previous two steps. To do this, first divide both of the original of this theorem (a² + b² = c²) by the square of the hypotenuse (a² / c² + b² / c² = 1), and then rewrite the resulting equality in this form: (a / c)² + (b / c )² = 1.

Replace in the resulting expression the ratio of the lengths of the legs and the hypotenuse with trigonometric functions, based on the formulas of the first and second steps: sin² (a) + cos² (a) \u003d 1. Express cosine from the resulting equality: cos(a) = √(1 - sin²(a)). This problem can be solved in a general way.

If, in addition to the general, you need to get a numerical result, use, for example, the calculator built into the Windows operating system. A link to its launch in the "Standard" subsection of the "All Programs" section of the OS menu. This link is worded concisely - "Calculator". To be able to calculate trigonometric functions from this program, turn on its "engineering" interface - press the key combination Alt + 2.

Enter the value of the sine of the angle in the conditions and click on the interface button with the designation x² - this will square the original value. Then type *-1 on the keyboard, press Enter, type +1 and press Enter again - in this way you will subtract the square of the sine from the unit. Click on the radical icon key to extract the square and get the final result.

One of the fundamental foundations of the exact sciences is the concept of trigonometric functions. They define simple relationships between the sides of a right triangle. The sine belongs to the family of these functions. Finding it, knowing the angle, can be done in a large number of ways, including experimental, computational methods, as well as the use of reference information.

You will need

• - calculator;
• - a computer;
• - spreadsheets;
• - bradys tables;
• - paper;
• - pencil.

Instruction

Use with the sine function to get the desired values ​​based on knowing the angle. Even the simplest ones have similar functionality today. In this case, calculations are made with a very high degree of accuracy (usually up to eight or more decimal places).

Apply software, which is a spreadsheet environment running on personal computer. Examples of such applications are Microsoft Office Excel and OpenOffice.org Calc. Enter in any cell a formula consisting of calling the sine function with the desired argument. Press Enter. The desired value will be displayed in the cell. The advantage of spreadsheets is the ability to quickly calculate function values ​​for a large set of arguments.

Find out the approximate value of the sine of the angle from the Bradys tables, if available. Their disadvantage is the accuracy of the values, which is limited to four decimal places.

Find the approximate value of the sine of the angle by making geometric constructions. Draw a line on a piece of paper. Using a protractor, set aside from it the angle whose sine you want to find. Draw another line that intersects the first one at some point. Perpendicular to the first segment, draw a straight line that intersects two existing segments. You get a right triangle. Measure the length of its hypotenuse and the leg opposite the angle constructed with the protractor. Divide the second value by the first. This will be the desired value.

Calculate the sine of an angle using the Taylor series expansion. If the angle value is in degrees, convert it to radians. Use a formula like this: sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + (x^9)/9! - ... To increase the speed of calculations, write down the current value of the numerator and denominator of the last member of the series, calculating the next value based on the previous one. Increase the length of the row for a more accurate value.

This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, and the cosine is the ratio of the adjacent leg to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used not only in right triangles. To find the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the cosine and sine theorem.

The cosine theorem is quite simple: "The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them."

There are two interpretations of the sine theorem: small and extended. According to the small: "In a triangle, the angles are proportional to the opposite sides." This theorem is often extended due to the property of the circle circumscribed about a triangle: "In a triangle, the angles are proportional to opposite sides, and their ratio is equal to the diameter of the circumscribed circle."

Derivatives

A derivative is a mathematical tool that shows how quickly a function changes with respect to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values ​​\u200b\u200bof the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the derivative of the cosine is the sine, but with a minus sign.

Application in mathematics

Especially often, sines and cosines are used in solving right triangles and problems related to them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking complex shapes and objects into "simple" triangles. Engineers and, often dealing with calculations of aspect ratios and degree measures, spent a lot of time and effort calculating cosines and sines of non-table angles.

Then the tables of Bradis came to the rescue, containing thousands of values ​​​​of sines, cosines, tangents and cotangents of different angles. In Soviet times, some teachers forced their wards to memorize the pages of the Bradis tables.

Radian - the angular value of the arc, along the length equal to the radius or 57.295779513 ° degrees.

Degree (in geometry) - 1/360th of a circle or 1/90th of a right angle.

π = 3.141592653589793238462… (approximate value of pi).

Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.

Angle x (in degrees)	0°	30°	45°	60°	90°	120°	135°	150°	180°	210°	225°	240°	270°	300°	315°	330°	360°
Angle x (in radians)	0	π/6	π/4	π/3	π/2	2 x π/3	3xπ/4	5xπ/6	π	7xπ/6	5xπ/4	4xπ/3	3xπ/2	5xπ/3	7xπ/4	11xπ/6	2xπ
cos x	1	√3/2 (0,8660)	√2/2 (0,7071)	1/2 (0,5)	0	-1/2 (-0,5)	-√2/2 (-0,7071)	-√3/2 (-0,8660)	-1	-√3/2 (-0,8660)	-√2/2 (-0,7071)	-1/2 (-0,5)	0	1/2 (0,5)	√2/2 (0,7071)	√3/2 (0,8660)	1

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. So

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.