Trigonometry formulas cos2x. All trigonometry formulas. Methods for solving trigonometric equations

The basic formulas of trigonometry are formulas that establish relationships between basic trigonometric functions. Sine, cosine, tangent and cotangent are interconnected by many relationships. Below we give the main trigonometric formulas, and for convenience we group them according to their purpose. Using these formulas, you can solve almost any problem from the standard trigonometry course. We note right away that only the formulas themselves are given below, and not their derivation, to which separate articles will be devoted.

Basic identities of trigonometry

Trigonometric identities give a relationship between the sine, cosine, tangent, and cotangent of one angle, allowing one function to be expressed in terms of another.

Trigonometric identities

sin 2 a + cos 2 a = 1 tg α = sin α cos α , ctg α = cos α sin α tg α ctg α = 1 tg 2 α + 1 = 1 cos 2 α , ctg 2 α + 1 = 1 sin 2α

These identities follow directly from the definitions of the unit circle, sine (sin), cosine (cos), tangent (tg), and cotangent (ctg).

Cast formulas

Casting formulas allow you to move from working with arbitrary and arbitrarily large angles to working with angles ranging from 0 to 90 degrees.

Cast formulas

sin α + 2 π z = sin α , cos α + 2 π z = cos α tg α + 2 π z = tg α , ctg α + 2 π z = ctg α sin - α + 2 π z = - sin α , cos - α + 2 π z = cos α tg - α + 2 π z = - tg α , ctg - α + 2 π z = - ctg α sin π 2 + α + 2 π z = cos α , cos π 2 + α + 2 π z = - sin α tg π 2 + α + 2 π z = - ctg α , ctg π 2 + α + 2 π z = - tg α sin π 2 - α + 2 π z = cos α , cos π 2 - α + 2 π z = sin α tg π 2 - α + 2 π z = ctg α , ctg π 2 - α + 2 π z = tg α sin π + α + 2 π z = - sin α , cos π + α + 2 π z = - cos α tg π + α + 2 π z = tg α , ctg π + α + 2 π z = ctg α sin π - α + 2 π z = sin α , cos π - α + 2 π z = - cos α tg π - α + 2 π z = - tg α , ctg π - α + 2 π z = - ctg α sin 3 π 2 + α + 2 π z = - cos α , cos 3 π 2 + α + 2 π z = sin α tg 3 π 2 + α + 2 π z = - ctg α , ctg 3 π 2 + α + 2 π z = - tg α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α tg 3 π 2 - α + 2 π z = ctg α , ctg 3 π 2 - α + 2 π z = tg α

The reduction formulas are a consequence of the periodicity of trigonometric functions.

Trigonometric addition formulas

The addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of angles in terms of the trigonometric functions of these angles.

Trigonometric addition formulas

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 tg α ± β = tg α ± tg β 1 ± tg α tg β ctg α ± β = - 1 ± ctg α ctg β ctg α ± ctg β

Based on the addition formulas, trigonometric formulas for a multiple angle are derived.

Multiple angle formulas: double, triple, etc.

Double and triple angle formulas

sin 2 α \u003d 2 sin α cos α cos 2 α \u003d cos 2 α - sin 2 α, cos 2 α \u003d 1 - 2 sin 2 α, cos 2 α \u003d 2 cos 2 α - 1 tg 2 α \u003d 2 tg α 1 - tg 2 α with tg 2 α \u003d with tg 2 α - 1 2 with tg α sin 3 α \u003d 3 sin α cos 2 α - sin 3 α, sin 3 α \u003d 3 sin α - 4 sin 3 α cos 3 α = cos 3 α - 3 sin 2 α cos α , cos 3 α = - 3 cos α + 4 cos 3 α tg 3 α = 3 tg α - tg 3 α 1 - 3 tg 2 α ctg 3 α = ctg 3 α - 3 ctg α 3 ctg 2 α - 1

Half Angle Formulas

The half angle formulas in trigonometry are a consequence of the double angle formulas and express the relationship between the basic functions of the half angle and the cosine of the whole angle.

Half Angle Formulas

sin 2 α 2 = 1 - cos α 2 cos 2 α 2 = 1 + cos α 2 t g 2 α 2 = 1 - cos α 1 + cos α c t g 2 α 2 = 1 + cos α 1 - cos α

Reduction Formulas

Reduction Formulas

sin 2 α = 1 - cos 2 α 2 cos 2 α = 1 + cos 2 α 2 sin 3 α = 3 sin α - sin 3 α 4 cos 3 α = 3 cos α + cos 3 α 4 sin 4 α = 3 - 4 cos 2 α + cos 4 α 8 cos 4 α = 3 + 4 cos 2 α + cos 4 α 8

Often, in calculations, it is inconvenient to operate with cumbersome powers. Degree reduction formulas allow you to reduce the degree of a trigonometric function from arbitrarily large to the first. Here is their general view:

General form of reduction formulas

for even n

sin n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 (- 1) n 2 - k C kn cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C kn cos ((n - 2 k) α)

for odd n

sin n α = 1 2 n - 1 ∑ k = 0 n - 1 2 (- 1) n - 1 2 - k C kn sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C kn cos ((n - 2 k) α)

Sum and difference of trigonometric functions

The difference and sum of trigonometric functions can be represented as a product. Factoring the differences of sines and cosines is very convenient to use when solving trigonometric equations and simplifying expressions.

Sum and difference of trigonometric functions

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 cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β \u003d - 2 sin α + β 2 sin α - β 2, cos α - cos β \u003d 2 sin α + β 2 sin β - α 2

Product of trigonometric functions

If the formulas for the sum and difference of functions allow you to go to their product, then the formulas for the product of trigonometric functions carry out the reverse transition - from the product to the sum. Formulas for the product of sines, cosines and sine by cosine are considered.

Formulas for the product of trigonometric functions

sin α sin β = 1 2 (cos (α - β) - cos (α + β)) cos α cos β = 1 2 (cos (α - β) + cos (α + β)) sin α cos β = 1 2 (sin (α - β) + sin (α + β))

Universal trigonometric substitution

All basic trigonometric functions - sine, cosine, tangent and cotangent - can be expressed in terms of the tangent of a half angle.

Universal trigonometric substitution

sin α = 2 tg α 2 1 + tg 2 α 2 cos α = 1 - tg 2 α 2 1 + tg 2 α 2 tg α = 2 tg α 2 1 - tg 2 α 2 ctg α = 1 - tg 2 α 2 2tgα 2

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

When performing trigonometric transformations, follow these tips:

1. Do not try to immediately come up with a scheme for solving an example from start to finish.
2. Don't try to convert the whole example at once. Move forward in small steps.
3. Remember that in addition to trigonometric formulas in trigonometry, you can still apply all the fair algebraic transformations (bracketing, reducing fractions, abbreviated multiplication formulas, and so on).
4. Believe that everything will be fine.

Basic trigonometric formulas

Most formulas in trigonometry are often applied both from right to left and from left to right, so you need to learn these formulas so well that you can easily apply some formula in both directions. To begin with, we write down the definitions of trigonometric functions. Let there be a right triangle:

Then, the definition of sine is:

Definition of cosine:

Definition of tangent:

Definition of cotangent:

Basic trigonometric identity:

The simplest corollaries from the basic trigonometric identity:

Double angle formulas. Sine of a double angle:

Cosine of a double angle:

Double angle tangent:

Double angle cotangent:

Additional trigonometric formulas

Trigonometric addition formulas. Sine of sum:

Sine of difference:

Cosine of the sum:

Cosine of difference:

Tangent of the sum:

Difference tangent:

Cotangent of the sum:

Difference cotangent:

Trigonometric formulas for converting a sum to a product. The sum of the sines:

Sine Difference:

Sum of cosines:

Cosine difference:

sum of tangents:

Tangent difference:

Sum of cotangents:

Cotangent difference:

Trigonometric formulas for converting a product into a sum. The product of sines:

The product of sine and cosine:

Product of cosines:

Degree reduction formulas.

Half Angle Formulas.

Trigonometric reduction formulas

The cosine function is called cofunction sine function and vice versa. Similarly, the functions tangent and cotangent are cofunctions. The reduction formulas can be formulated as the following rule:

• If in the reduction formula the angle is subtracted (added) from 90 degrees or 270 degrees, then the reducible function changes to a cofunction;
• If in the reduction formula the angle is subtracted (added) from 180 degrees or 360 degrees, then the name of the reduced function is preserved;
• In this case, the reduced function is preceded by the sign that the reduced (i.e., original) function has in the corresponding quarter, if we consider the subtracted (added) angle to be acute.

Cast formulas are given in the form of a table:

By trigonometric circle it is easy to determine tabular values ​​of trigonometric functions:

Trigonometric equations

To solve a certain trigonometric equation, it must be reduced to one of the simplest trigonometric equations, which will be discussed below. For this:

• You can apply the trigonometric formulas above. In this case, you do not need to try to convert the entire example at once, but you need to move forward in small steps.
• We must not forget about the possibility of transforming some expression with the help of algebraic methods, i.e. for example, put something out of the bracket or, conversely, open the brackets, reduce the fraction, apply the abbreviated multiplication formula, reduce fractions to a common denominator, and so on.
• When solving trigonometric equations, you can apply grouping method. It must be remembered that in order for the product of several factors to be equal to zero, it is enough that any of them be equal to zero, and the rest existed.
• Applying variable replacement method, as usual, the equation after the introduction of the replacement should become simpler and not contain the original variable. You also need to remember to do the reverse substitution.
• Remember that homogeneous equations often occur in trigonometry as well.
• When opening modules or solving irrational equations with trigonometric functions, one must remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
• Remember about the ODZ (in trigonometric equations, the restrictions on the ODZ basically boil down to the fact that you cannot divide by zero, but do not forget about other restrictions, especially about the positivity of expressions in rational powers and under roots of even degrees). Also remember that sine and cosine values ​​can only lie between minus one and plus one, inclusive.

The main thing is, if you don’t know what to do, do at least something, while the main thing is to use trigonometric formulas correctly. If what you get is getting better and better, then continue with the solution, and if it gets worse, then go back to the beginning and try applying other formulas, so do until you stumble upon the correct solution.

Formulas for solving the simplest trigonometric equations. For the sine, there are two equivalent forms of writing the solution:

For other trigonometric functions, the notation is unique. For cosine:

For tangent:

For cotangent:

Solution of trigonometric equations in some special cases:

Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own surname. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points, as well as responsible study of the final training tests, will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you, as it seems to you, have found an error in the training materials, then please write about it by e-mail (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

This is the last and most important lesson needed to solve problems B11. We already know how to convert angles from radians to degrees (see lesson " Radian and degree measure of an angle”), and we also know how to determine the sign of the trigonometric function, focusing on the coordinate quarters (see the lesson “ Signs of trigonometric functions »).

The matter remains small: to calculate the value of the function itself - the very number that is written in the answer. Here the basic trigonometric identity comes to the rescue.

Basic trigonometric identity. For any angle α, the statement is true:

sin 2 α + cos 2 α = 1.

This formula relates the sine and cosine of one angle. Now, knowing the sine, we can easily find the cosine - and vice versa. It is enough to take the square root:

Notice the "±" sign in front of the roots. The fact is that from the basic trigonometric identity it is not clear what the original sine and cosine were: positive or negative. After all, squaring is an even function that "burns" all the minuses (if any).

That is why in all B11 tasks that are found in the USE in mathematics, there are necessarily additional conditions that help get rid of uncertainty with signs. Usually this is an indication of the coordinate quarter by which the sign can be determined.

An attentive reader will surely ask: “What about the tangent and cotangent?” It is impossible to directly calculate these functions from the above formulas. However, there are important corollaries from the basic trigonometric identity that already contain tangents and cotangents. Namely:

An important corollary: for any angle α, the basic trigonometric identity can be rewritten as follows:

These equations are easily deduced from the basic identity - it is enough to divide both sides by cos 2 α (to get the tangent) or by sin 2 α (for the cotangent).

Let's look at all this with specific examples. The following are actual B11 problems taken from the 2012 Mathematics USE trials.

We know the cosine, but we don't know the sine. The main trigonometric identity (in its "pure" form) connects just these functions, so we will work with it. We have:

sin 2 α + cos 2 α = 1 ⇒ sin 2 α + 99/100 = 1 ⇒ sin 2 α = 1/100 ⇒ sin α = ±1/10 = ±0.1.

To solve the problem, it remains to find the sign of the sine. Since the angle α ∈ (π /2; π ), then in degree measure it is written as follows: α ∈ (90°; 180°).

Therefore, the angle α lies in the II coordinate quarter - all the sines there are positive. Therefore sin α = 0.1.

So, we know the sine, but we need to find the cosine. Both of these functions are in the basic trigonometric identity. We substitute:

sin 2 α + cos 2 α = 1 ⇒ 3/4 + cos 2 α = 1 ⇒ cos 2 α = 1/4 ⇒ cos α = ±1/2 = ±0.5.

It remains to deal with the sign in front of the fraction. What to choose: plus or minus? By condition, the angle α belongs to the interval (π 3π /2). Let's convert the angles from radian measure to degree measure - we get: α ∈ (180°; 270°).

Obviously, this is the III coordinate quarter, where all cosines are negative. Therefore cosα = −0.5.

Task. Find tg α if you know the following:

Tangent and cosine are related by an equation following from the basic trigonometric identity:

We get: tg α = ±3. The sign of the tangent is determined by the angle α. It is known that α ∈ (3π /2; 2π ). Let's convert the angles from the radian measure to the degree measure - we get α ∈ (270°; 360°).

Obviously, this is the IV coordinate quarter, where all tangents are negative. Therefore, tgα = −3.

Task. Find cos α if you know the following:

Again, the sine is known and the cosine is unknown. We write down the main trigonometric identity:

sin 2 α + cos 2 α = 1 ⇒ 0.64 + cos 2 α = 1 ⇒ cos 2 α = 0.36 ⇒ cos α = ±0.6.

The sign is determined by the angle. We have: α ∈ (3π /2; 2π ). Let's convert the angles from degrees to radians: α ∈ (270°; 360°) is the IV coordinate quarter, the cosines are positive there. Therefore, cos α = 0.6.

Task. Find sin α if you know the following:

Let's write a formula that follows from the basic trigonometric identity and directly connects the sine and cotangent:

From here we get that sin 2 α = 1/25, i.e. sin α = ±1/5 = ±0.2. It is known that the angle α ∈ (0; π /2). In degrees, this is written as follows: α ∈ (0°; 90°) - I coordinate quarter.

So, the angle is in the I coordinate quarter - all trigonometric functions are positive there, therefore sin α \u003d 0.2.