Basic trigonometry formulas are formulas that establish connections between basic trigonometric functions. Sine, cosine, tangent and cotangent are interconnected by many relationships. Below we present the main trigonometric formulas, and for convenience we will group them by purpose. Using these formulas you can solve almost any problem from a standard trigonometry course. Let us immediately note that below are only the formulas themselves, and not their conclusion, which will be discussed in separate articles.

Basic identities of trigonometry

Trigonometric identities provide 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 t g α = sin α cos α , c t g α = cos α sin α t g α c t g α = 1 t g 2 α + 1 = 1 cos 2 α , c t g 2 α + 1 = 1 sin 2 α

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

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.

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

Reduction formulas are a consequence of periodicity trigonometric functions.

Trigonometric addition formulas

Addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of angles in terms of 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 t g α ± β = t g α ± t g β 1 ± t g α t g β c t g α ± β = - 1 ± c t g α c t g β c t g α ± c t g β

Based on addition formulas, trigonometric formulas for multiple angles are derived.

Formulas for multiple angles: double, triple, etc.

Double and triple angle formulas

sin 2 α = 2 · sin α · cos α cos 2 α = cos 2 α - sin 2 α , cos 2 α = 1 - 2 sin 2 α , cos 2 α = 2 cos 2 α - 1 t g 2 α = 2 · t g α 1 - t g 2 α with t g 2 α = with t g 2 α - 1 2 · with t g α sin 3 α = 3 sin α · cos 2 α - sin 3 α , sin 3 α = 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 t g 3 α = 3 t g α - t g 3 α 1 - 3 t g 2 α c t g 3 α = c t g 3 α - 3 c t g α 3 c t g 2 α - 1

Half angle formulas

Half-angle formulas in trigonometry are a consequence of double-angle formulas and express the relationship between the basic functions of a half-angle and the cosine of a 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

Degree reduction formulas

Degree 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

It is often inconvenient to work with cumbersome powers when making calculations. 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 view of the degree 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 k n · cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C k n 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 k n sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C k n 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 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 = - 2 sin α + β 2 sin α - β 2 , cos α - cos β = 2 sin α + β 2 sin β - α 2

Product of trigonometric functions

If the formulas for the sum and difference of functions allow one 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 t g α 2 1 + t g 2 α 2 cos α = 1 - t g 2 α 2 1 + t g 2 α 2 t g α = 2 t g α 2 1 - t g 2 α 2 c t g α = 1 - t g 2 α 2 2 t g α 2

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Basic formulas of trigonometry. Lesson No. 1

The number of formulas used in trigonometry is quite large (by “formulas” we do not mean definitions (for example, tgx=sinx/cosx), but identical equalities like sin2x=2sinxcosx). To make it easier to navigate this abundance of formulas and not tire students with meaningless cramming, it is necessary to highlight the most important ones among them. There are few of them - only three. All the others follow from these three formulas. This is the basic trigonometric identity and formulas for the sine and cosine of the sum and difference:

Sin 2 x+cos 2 x=1 (1)

Sin(x±y)=sinxcosy±sinycosx (2)

Cos(x±y)=cosxcosy±sinxsiny (3)

From these three formulas follow absolutely all the properties of sine and cosine (periodicity, period value, sine value 30 0 = π/6=1/2, etc.) From this point of view, in school curriculum A lot of formally unnecessary, redundant information is used. So, formulas “1-3” are the rulers of the trigonometric kingdom. Let's move on to the corollary formulas:

1) Sines and cosines of multiple angles

If we substitute the value x=y into (2) and (3), we get:

Sin2x=2sinxcosх; sin0=sinxcosx-sinxcosx=0

Cos2x=cos 2 x-sin 2 x; cos0=cos 2 x+sin 2 x=1

We deduced that sin0=0; cos0=1, without resorting to the geometric interpretation of sine and cosine. Similarly, by applying the "2-3" formulas twice, we can derive expressions for sin3x; cos3x; sin4x; cos4x, etc.

Sin3x = sin(2x+x) = sin2xcosx+sinxcos2x = 2sinxcos 2 x+sinx(cos 2 x-sin 2 x) = 2sinx(1-sin 2 x)+sinx(1-2sin 2 x) = 3sinx-4sin 3 x

Task for students: derive similar expressions for cos3x; sin4x; cos4x

2) Degree reduction formulas

Solve the inverse problem by expressing the powers of sine and cosine in terms of cosines and sines of multiple angles.

For example: cos2x=cos 2 x-sin 2 x=2cos 2 x-1, hence: cos 2 x=1/2+cos2x/2

Cos2x=cos 2 x-sin 2 x=1-2sin 2 x, hence: sin 2 x=1/2-cos2x/2

These formulas are used very often. To understand them better, I advise you to draw graphs of their left and right sides. The graphs of the squares of cosine and sine “wrap around” the graph of the straight line “y=1/2” (this is the average over many periods cos value 2 x and sin 2 x). In this case, the oscillation frequency doubles compared to the original (the period of the functions cos 2 x sin 2 x is equal to 2π /2=π), and the amplitude of the oscillations is halved (coefficient 1/2 before cos2x).

Problem: Express sin 3 x; cos 3 x; sin 4 x ; cos 4 x through cosines and sines of multiple angles.

3) Reduction formulas

They use the periodicity of trigonometric functions, allowing their values ​​to be calculated in any quarter of the trigonometric circle from the values ​​in the first quarter. Reduction formulas are very special cases of the “main” formulas (2-3). For example: cos(x+π/2)=cosxcos π/2-sinxsin π/2=cosx*0-sinx*1=sinx

So Cos(x+ π/2) =sinx

Task: derive reduction formulas for sin(x+ π/2); cos(x+ 3 π/2)

4) Formulas that convert the sum or difference of cosine and sine into a product and vice versa.

Let's write out the formula for the sine of the sum and difference of two angles:

Sin(x+y) = sinxcosy+sinycosx (1)

Sin(x-y) = sinxcosy-sinycosx (2)

Let's add the left and right sides of these equalities:

Sin(x+y) +sin(x-y) = sinxcosy +sinycosx +sinxcosy –sinycosx

Similar terms cancel, so:

Sin(x+y) +sin(x-y) = 2sinxcosy (*)

a) when reading (*) from right to left, we get:

Sinxcosy= 1/2(sin(x+y) + sin(x-y)) (4)

The product of the sines of two angles is equal to half the sum of the sines of the sum and the difference of these angles.

b) when reading (*) from left to right, it is convenient to denote:

x-y = c. From here we will find X And at through R And With, adding and subtracting the left and right sides of these two equalities:

x = (p+c)/2, y = (p-c)/2, substituting in (*) instead of (x+y) and (x-y) the derived new variables R And With, let’s imagine the sum of sines through the product:

sinp + sinc =2sin(p+c)/2cos(p-c)/2 (5)

So, a direct consequence of the basic formula for the sine of the sum and the difference of angles turns out to be two new relations (4) and (5).

c) now, instead of adding the left and right sides of equalities (1) and (2), we will subtract them from each other:

sin(x+y) – sin(x-y) = 2sinycosx (6)

Reading this identity from right to left leads to a formula similar to (4), which turns out to be uninteresting, because we already know how to decompose the products of sine and cosine into a sum of sines (see (4)). Reading (6) from left to right gives a formula that collapses the difference of sines into a product:

sinp – sinc = 2sin((p-c)/2) * cos((p+c)/2) (7)

So, from one fundamental identity sin (x±y) = sinxcosy±sinycosx, we got three new ones (4), (5), (7).

Similar work done with another fundamental identity cos (x±y) = cosxcosy±sinxsiny leads to four new ones:

Cosxcosy = ½ (cos(x+y) + cos(x-y)); cosp + cosc ​​= 2cos((p+c)/2)cos((p-c)/2);

Sinxsiny = ½ (cos(x-y) – cos(x+y)); cosp-cosc = -2sin((p-c)/2)sin((p+c)/2)

Task: convert the sum of sine and cosine into a product:

Sinx +cozy = ? Solution: if you try not to derive the formula, but immediately look at the answer in some table of trigonometric formulas, then you may not find a ready-made result. Students should understand that there is no need to memorize and enter into the table another formula for sinx + cosy = ..., since any cosine can be represented as a sine and, conversely, using reduction formulas, for example: sinx = cos (π/2 – x), cozy = sin (π/2 – y). Therefore: sinx+cosy = sinx + sin (π/2 – y) = 2sin ((x+π/2 – y)/2)cos((x - π/2 + y)/2.