Table of conversion of trigonometric functions. Basic trigonometry formulas

Trigonometry is one of the branches of mathematics, the focus of which is 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 use found expressions for defining a double argument. For example, by resorting to the corresponding equation, one can easily find out double angle sinus.

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 given angles 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 sine and cosine of a 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 to calculate "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!

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Double angle formulas are used to express the sines, cosines, tangents, cotangents of an angle with a value of 2 α using the trigonometric functions of the angle α . This article will introduce all double angle formulas with proofs. Examples of application of formulas will be considered. In the final part, the formulas for the triple, quadruple angles will be shown.

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List of double angle formulas

To convert double angle formulas, remember that angles in trigonometry have the form n α notation, where n is natural number, the value of the expression is written without brackets. Thus, sin n α is considered to have the same meaning as sin (n α) . With the notation sin n α we have a similar notation (sin α) n . The use of the record is applicable to everyone trigonometric functions with powers of n.

The following are the double 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 c t g 2 α - c t g 2 α - 1 2 c t g α

Note that these sin and cos formulas are applicable with any value of the angle α. The formula for the tangent of a double angle is valid for any value of α, where t g 2 α makes sense, that is, α ≠ π 4 + π 2 · z, z is any integer. The cotangent of a double angle exists for any α , where c t g 2 α is defined on α ≠ π 2 · z .

The cosine of a double angle has a triple notation of a double angle. All of them are applicable.

Proof of double angle formulas

The proof of the formulas originates from the addition formulas. We apply the formulas for the sine of the sum:

sin (α + β) = sin α cos β + cos α sin β and the cosine of the sum cos (α + β) = cos α cos β - sin α sin β. Suppose that β = α , then we get that

sin (α + α) = sin α cos α + cos α sin α = 2 sin α cos α and cos (α + α) = cos α cos α - sin α sin α = cos 2 α - sin2α

Thus, the formulas for the sine and cosine of the double angle sin 2 α \u003d 2 sin α cos α and cos 2 α \u003d cos 2 α - sin 2 α are proved.

Rest cos formulas 2 α \u003d 1 - 2 sin 2 α and cos 2 α \u003d 2 cos 2 α - 1 lead to the form cos 2 α \u003d cos 2 α \u003d cos 2 α - sin 2 α, when replacing 1 with the sum of squares according to the main identity sin 2 α + cos 2 α = 1 . We get that sin 2 α + cos 2 α = 1. So 1 - 2 sin 2 α \u003d sin 2 α + cos 2 α - 2 sin 2 α \u003d cos 2 α - sin 2 α and 2 cos 2 α - 1 \u003d 2 cos 2 α - (sin 2 α + cos 2 α) \u003d cos 2 α - sin 2 α.

To prove the formulas for the double angle of tangent and cotangent, we apply the equalities t g 2 α \u003d sin 2 α cos 2 α and c t g 2 α \u003d cos 2 α sin 2 α. After the transformation, we get that t g 2 α \u003d sin 2 α cos 2 α \u003d 2 sin α cos α cos 2 α - sin 2 α and c t g 2 α \u003d cos 2 α sin 2 α \u003d cos 2 α - sin 2 α 2 · sin α · cos α . Divide the expression by cos 2 α where cos 2 α ≠ 0 with any value of α when t g α is defined. Divide another expression by sin 2 α , where sin 2 α ≠ 0 with any values ​​of α , when c t g 2 α makes sense. To prove the double angle formula for tangent and cotangent, we substitute and get:

- surely there will be tasks in trigonometry. Trigonometry is often disliked for having to cram a huge amount of difficult formulas teeming with sines, cosines, tangents and cotangents. The site already once gave advice on how to remember a forgotten formula, using the example of the Euler and Peel formulas.

And in this article we will try to show that it is enough to firmly know only five of the simplest trigonometric formulas, and about the rest to have a general idea and display them as you go. It's like with DNA: the complete drawings of a finished living being are not stored in the molecule. It contains, rather, instructions for assembling it from the available amino acids. So it is in trigonometry, knowing some general principles, we will get all the necessary formulas from a small set of those that must be kept in mind.

We will rely on the following formulas:

From the formulas for the sine and cosine of the sums, knowing that the cosine function is even and that the sine function is odd, substituting -b for b, we obtain formulas for the differences:

1. Sine of difference: sin(a-b) = sinacos(-b)+cosasin(-b) = sinacosb-cosasinb
2. cosine difference: cos(a-b) = cosacos(-b)-sinasin(-b) = cosacosb+sinasinb

Putting a \u003d b into the same formulas, we obtain the formulas for the sine and cosine of double angles:

1. Sine of a double angle: sin2a = sin(a+a) = sinacosa+cosasina = 2sinacosa
2. Cosine of a double angle: cos2a = cos(a+a) = cosacosa-sinasina = cos2a-sin2a

The formulas for other multiple angles are obtained similarly:

1. Sine of a triple angle: sin3a = sin(2a+a) = sin2acosa+cos2asina = (2sinacosa)cosa+(cos2a-sin2a)sina = 2sinacos2a+sinacos2a-sin 3 a = 3 sinacos2a-sin 3 a = 3 sina(1-sin2a)-sin 3 a = 3 sina-4sin 3a
2. Cosine of a triple angle: cos3a = cos(2a+a) = cos2acosa-sin2asina = (cos2a-sin2a)cosa-(2sinacosa)sina = cos 3a- sin2acosa-2sin2acosa = cos 3a-3 sin2acosa = cos 3 a-3(1- cos2a)cosa = 4cos 3a-3 cosa

Before moving on, let's consider one problem.
Given: the angle is acute.
Find its cosine if
Solution given by one student:
Because , then sina= 3,a cosa = 4.
(From mathematical humor)

So, the definition of tangent connects this function with both sine and cosine. But you can get a formula that gives the connection of the tangent only with the cosine. To derive it, we take the main trigonometric identity: sin 2 a+cos 2 a= 1 and divide it by cos 2 a. We get:

So the solution to this problem would be:

(Because the angle is acute, the + sign is taken when extracting the root)

The formula for the tangent of the sum is another one that is hard to remember. Let's output it like this:

immediately output and

From the cosine formula for a double angle, you can get the sine and cosine formulas for a half angle. To do this, to the left side of the double angle cosine formula:
cos2 a = cos 2 a-sin 2 a
we add a unit, and to the right - a trigonometric unit, i.e. sum of squares of sine and cosine.
cos2a+1 = cos2a-sin2a+cos2a+sin2a
2cos 2 a = cos2 a+1
expressing cosa through cos2 a and performing a change of variables, we get:

The sign is taken depending on the quadrant.

Similarly, subtracting one from the left side of the equality, and the sum of the squares of the sine and cosine from the right side, we get:
cos2a-1 = cos2a-sin2a-cos2a-sin2a
2sin 2 a = 1-cos2 a

And finally, to convert the sum of trigonometric functions into a product, we use the following trick. Suppose we need to represent the sum of sines as a product sina+sinb. Let's introduce variables x and y such that a = x+y, b+x-y. Then
sina+sinb = sin(x+y)+ sin(x-y) = sin x cos y+ cos x sin y+ sin x cos y- cos x sin y=2 sin x cos y. Let us now express x and y in terms of a and b.

Since a = x+y, b = x-y, then . So

You can withdraw immediately

1. Partition formula products of sine and cosine in amount: sinacosb = 0.5(sin(a+b)+sin(a-b))

We recommend that you practice and derive formulas for converting the product of the difference of sines and the sum and difference of cosines into a product, as well as for splitting the products of sines and cosines into a sum. Having done these exercises, you will thoroughly master the skill of deriving trigonometric formulas and will not get lost even in the most difficult control, olympiad or testing.