The simplest trigonometric identities
The quotient of dividing the sine of the angle alpha by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of the angle alpha by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of squares of cosine and sine.
The sum of the unit and the tangent of the angle is equal to the ratio of the unit to the square of the cosine of this angle (Formula 5)
The unit plus the cotangent of the angle is equal to the quotient of dividing the unit by the sine square of this angle (Formula 6)
The product of the tangent and the cotangent of the same angle is equal to one (Formula 7).
Converting negative angles of trigonometric functions (even and odd)
In order to get rid of the negative value of the degree measure of the angle when calculating the sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of even or odd trigonometric functions.
As seen, cosine and secant is even function
, sine, tangent and cotangent are odd functions.
Sinus negative angle equals negative value the sine of the same positive angle(minus sine alpha).
The cosine "minus alpha" will give the same value as the cosine of the angle alpha.
Tangent minus alpha is equal to minus tangent alpha.
Double angle reduction formulas (sine, cosine, tangent and cotangent of a double angle)
If it is necessary to bisect the angle, or vice versa, go from double angle to single, you can use the following trigonometric identities:
Double Angle Conversion (double angle sine, double angle cosine and double angle tangent) into a single one occurs according to the following rules:
Sine of a double angle is equal to twice the product of the sine and the cosine of a single angle
Cosine of a double angle is equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle
Cosine of a double angle equal to twice the square of the cosine of a single angle minus one
Cosine of a double angle equals one minus the double sine square of a single angle
Double angle tangent is equal to a fraction whose numerator is twice the tangent of a single angle, and whose denominator is equal to one minus the tangent of the square of a single angle.
Double angle cotangent is equal to a fraction whose numerator is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle
Universal Trigonometric Substitution Formulas
The conversion formulas below can be useful when you need to divide the argument of the trigonometric function (sin α, cos α, tg α) by two and bring the expression to the value of half the angle. From the value of α we get α/2 .These formulas are called formulas of the universal trigonometric substitution. Their value lies in the fact that the trigonometric expression with their help is reduced to the expression of the tangent of half an angle, no matter what trigonometric functions ( sin cos tg ctg) were originally in the expression. After that, the equation with the tangent of half an angle is much easier to solve.
Trigonometric half-angle transformation identities
Formulas below trigonometric transformation half the value of the angle to its integer value.The value of the argument of the trigonometric function α/2 is reduced to the value of the argument of the trigonometric function α.
Trigonometric formulas for adding angles
cos (α - β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + sin β cos α
sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β
Tangent and cotangent of the sum of angles alpha and beta can be converted according to the following rules for converting trigonometric functions:
Tangent of sum of angles is equal to a fraction whose numerator is the sum of the tangent of the first and the tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.
Angle difference tangent is equal to a fraction, the numerator of which is equal to the difference between the tangent of the reduced angle and the tangent of the angle to be subtracted, and the denominator is one plus the product of the tangents of these angles.
Cotangent of sum of angles is equal to a fraction whose numerator is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.
Cotangent of angle difference is equal to a fraction whose numerator is the product of the cotangents of these angles minus one, and the denominator is equal to the sum cotangents of these angles.
These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If it is represented as tg (45 + 60), then you can use the given identical transformations of the tangent of the sum of the angles, after which you simply substitute the tabular values of the tangent of 45 and the tangent of 60 degrees.
Formulas for converting the sum or difference of trigonometric functions
Expressions representing the sum of the form sin α + sin β can be converted using the following formulas:
Triple angle formulas - convert sin3α cos3α tg3α to sinα cosα tgα
Sometimes it is necessary to convert the triple value of the angle so that the angle α becomes the argument of the trigonometric function instead of 3α.In this case, you can use the formulas (identities) for the transformation of the triple angle:
Formulas for transforming the product of trigonometric functions
If it becomes necessary to convert the product of sines of different angles of cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:
In this case, the product of the sine, cosine or tangent functions of different angles will be converted to a sum or difference.
Formulas for reducing trigonometric functions
You need to use the cast table as follows. In the line, select the function that interests us. The column is an angle. For example, the sine of the angle (α+90) at the intersection of the first row and the first column, we find out that sin (α+90) = cos α .
AT identical transformations trigonometric expressions the following algebraic tricks can be used: adding and subtracting identical terms; taking the common factor out of brackets; multiplication and division by the same value; application of abbreviated multiplication formulas; selection full square; decomposition square trinomial for multipliers; introduction of new variables to simplify transformations.
When converting trigonometric expressions containing fractions, you can use the properties of proportion, reduce fractions, or convert fractions to common denominator. In addition, you can use the selection of the integer part of the fraction, multiplying the numerator and denominator of the fraction by the same value, and also, if possible, take into account the uniformity of the numerator or denominator. If necessary, you can represent a fraction as a sum or difference of several simpler fractions.
In addition, applying all the necessary methods for converting trigonometric expressions, it is necessary to constantly take into account the area allowed values converted expressions.
Let's look at a few examples.
Example 1
Calculate A = (sin (2x - π) cos (3π - x) + sin (2x - 9π/2) cos (x + π/2)) 2 + (cos (x - π/2) cos ( 2x – 7π/2) +
+
sin (3π/2 - x) sin (2x -5π/2)) 2
Solution.
It follows from the reduction formulas:
sin (2x - π) \u003d -sin 2x; cos (3π - x) \u003d -cos x;
sin (2x - 9π / 2) \u003d -cos 2x; cos (x + π/2) = -sin x;
cos (x - π / 2) \u003d sin x; cos (2x - 7π/2) = -sin 2x;
sin (3π / 2 - x) \u003d -cos x; sin (2x - 5π / 2) \u003d -cos 2x.
Whence, by virtue of the formulas for the addition of arguments and the basic trigonometric identity, we obtain
A \u003d (sin 2x cos x + cos 2x sin x) 2 + (-sin x sin 2x + cos x cos 2x) 2 \u003d sin 2 (2x + x) + cos 2 (x + 2x) \u003d
= sin 2 3x + cos 2 3x = 1
Answer: 1.
Example 2
Convert the expression M = cos α + cos (α + β) cos γ + cos β – sin (α + β) sin γ + cos γ into a product.
Solution.
From the formulas for the addition of arguments and the formulas for converting the sum of trigonometric functions into a product, after the appropriate grouping, we have
М = (cos (α + β) cos γ - sin (α + β) sin γ) + cos α + (cos β + cos γ) =
2cos ((β + γ)/2) cos ((β – γ)/2) + (cos α + cos (α + β + γ)) =
2cos ((β + γ)/2) cos ((β – γ)/2) + 2cos (α + (β + γ)/2) cos ((β + γ)/2)) =
2cos ((β + γ)/2) (cos ((β – γ)/2) + cos (α + (β + γ)/2)) =
2cos ((β + γ)/2) 2cos ((β – γ)/2 + α + (β + γ)/2)/2) cos ((β – γ)/2) – (α + ( β + γ)/2)/2) =
4cos ((β + γ)/2) cos ((α + β)/2) cos ((α + γ)/2).
Answer: М = 4cos ((α + β)/2) cos ((α + γ)/2) cos ((β + γ)/2).
Example 3.
Show that the expression A \u003d cos 2 (x + π / 6) - cos (x + π / 6) cos (x - π / 6) + cos 2 (x - π / 6) takes for all x from R one and the same value. Find this value.
Solution.
We present two methods for solving this problem. Applying the first method, by isolating the full square and using the corresponding basic trigonometric formulas, we obtain
A \u003d (cos (x + π / 6) - cos (x - π / 6)) 2 + cos (x - π / 6) cos (x - π / 6) \u003d
4sin 2 x sin 2 π/6 + 1/2(cos 2x + cos π/3) =
Sin 2 x + 1/2 cos 2x + 1/4 = 1/2 (1 - cos 2x) + 1/2 cos 2x + 1/4 = 3/4.
Solving the problem in the second way, consider A as a function of x from R and calculate its derivative. After transformations, we get
А´ \u003d -2cos (x + π/6) sin (x + π/6) + (sin (x + π/6) cos (x - π/6) + cos (x + π/6) sin (x + π/6)) - 2cos (x - π/6) sin (x - π/6) =
Sin 2(x + π/6) + sin ((x + π/6) + (x - π/6)) - sin 2(x - π/6) =
Sin 2x - (sin (2x + π/3) + sin (2x - π/3)) =
Sin 2x - 2sin 2x cos π/3 = sin 2x - sin 2x ≡ 0.
Hence, by virtue of the criterion of constancy of a function differentiable on an interval, we conclude that
A(x) ≡ (0) = cos 2 π/6 - cos 2 π/6 + cos 2 π/6 = (√3/2) 2 = 3/4, x ∈ R. 
Answer: A = 3/4 for x € R.
The main methods of proving trigonometric identities are:
a) reduction of the left side of the identity to the right side by appropriate transformations;
b) reduction of the right side of the identity to the left;
in) reduction of the right and left parts of the identity to the same form;
G) reduction to zero of the difference between the left and right parts of the identity being proved.
Example 4
Check that cos 3x = -4cos x cos (x + π/3) cos (x + 2π/3).
Solution.
Transforming the right side of this identity according to the corresponding trigonometric formulas, we have
4cos x cos (x + π/3) cos (x + 2π/3) =
2cos x (cos ((x + π/3) + (x + 2π/3)) + cos ((x + π/3) – (x + 2π/3))) =
2cos x (cos (2x + π) + cos π/3) =
2cos x cos 2x - cos x = (cos 3x + cos x) - cos x = cos 3x.
The right side of the identity is reduced to the left side.
Example 5
Prove that sin 2 α + sin 2 β + sin 2 γ – 2cos α cos β cos γ = 2 if α, β, γ are interior angles of some triangle.
Solution.
Taking into account that α, β, γ are interior angles of some triangle, we obtain that
α + β + γ = π and hence γ = π – α – β.
sin 2 α + sin 2 β + sin 2 γ – 2cos α cos β cos γ =
Sin 2 α + sin 2 β + sin 2 (π - α - β) - 2cos α cos β cos (π - α - β) =
Sin 2 α + sin 2 β + sin 2 (α + β) + (cos (α + β) + cos (α - β) (cos (α + β) =
Sin 2 α + sin 2 β + (sin 2 (α + β) + cos 2 (α + β)) + cos (α - β) (cos (α + β) =
1/2 (1 - cos 2α) + ½ (1 - cos 2β) + 1 + 1/2 (cos 2α + cos 2β) = 2.
The original equality is proved.
Example 6
Prove that in order for one of the angles α, β, γ of the triangle to be equal to 60°, it is necessary and sufficient that sin 3α + sin 3β + sin 3γ = 0.
Solution.
The condition of this problem presupposes the proof of both necessity and sufficiency.
First we prove need.
It can be shown that
sin 3α + sin 3β + sin 3γ = -4cos (3α/2) cos (3β/2) cos (3γ/2).
Hence, taking into account that cos (3/2 60°) = cos 90° = 0, we obtain that if one of the angles α, β or γ is equal to 60°, then
cos (3α/2) cos (3β/2) cos (3γ/2) = 0 and hence sin 3α + sin 3β + sin 3γ = 0.
Let's prove now adequacy the specified condition.
If sin 3α + sin 3β + sin 3γ = 0, then cos (3α/2) cos (3β/2) cos (3γ/2) = 0, and therefore
either cos (3α/2) = 0, or cos (3β/2) = 0, or cos (3γ/2) = 0.
Consequently,
or 3α/2 = π/2 + πk, i.e. α = π/3 + 2πk/3, 
or 3β/2 = π/2 + πk, i.e. β = π/3 + 2πk/3,
or 3γ/2 = π/2 + πk,
those. γ = π/3 + 2πk/3, where k ϵ Z.
From the fact that α, β, γ are the angles of a triangle, we have
0 < α < π, 0 < β < π, 0 < γ < π.
Therefore, for α = π/3 + 2πk/3 or β = π/3 + 2πk/3 or
γ = π/3 + 2πk/3 out of all kϵZ only k = 0 fits.
Whence it follows that either α = π/3 = 60°, or β = π/3 = 60°, or γ = π/3 = 60°.
The assertion has been proven.
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Executed for all values of the argument (from the general scope).
Universal substitution formulas.
With these formulas, it is easy to turn any expression that contains various trigonometric functions of one argument into a rational expression of one function. tg (α /2):
Formulas for converting sums to products and products to sums.
Previously, the above formulas were used to simplify calculations. They calculated using logarithmic tables, and later - a slide rule, since logarithms are best suited for multiplying numbers. That is why each original expression was reduced to a form that would be convenient for logarithms, that is, to products, for example:
2 sin α sin b = cos (α - b) - cos (α + b);
2 cos α cos b = cos (α - b) + cos (α + b);
2 sin α cos b = sin (α - b) + sin (α + b).
where is the angle for which, in particular,
Formulas for the tangent and cotangent functions are easily obtained from the above.
Degree reduction formulas.
|
sin 2 α \u003d (1 - cos 2α) / 2; |
cos2α = (1 + cos2α)/2; |
|
sin 3α = (3 sinα -sin 3α )/4; |
cos 3 a = (3 cosα + cos 3α )/4. |
With the help of these formulas trigonometric equations are easily reduced to equations with lower powers. In the same way, lowering formulas are derived for more high degrees sin and cos.
Expression of trigonometric functions through one of them of the same argument.
The sign in front of the root depends on the quarter of the angle α .
