Basic trigonometric identities.
secα read: “secant alpha”. This is the reciprocal of cosine alpha.
cosecα read: “cosecant alpha.” This is the reciprocal of sine alpha.
Examples. Simplify the expression:
A) 1 – sin 2 α; b) cos 2 α – 1; V)(1 – cosα)(1+cosα); G) sin 2 αcosα – cosα; d) sin 2 α+1+cos 2 α;
e) sin 4 α+2sin 2 αcos 2 α+cos 4 α; and) tg 2 α – sin 2 αtg 2 α; h) ctg 2 αcos 2 α – ctg 2 α; And) cos 2 α+tg 2 αcos 2 α.
A) 1 – sin 2 α = cos 2 α according to the formula 1) ;
b) cos 2 α – 1 =- (1 – cos 2 α) = -sin 2 α also applied the formula 1) ;
V)(1 – cosα)(1+cosα) = 1 – cos 2 α = sin 2 α. First, we applied the formula for the difference of the squares of two expressions: (a – b)(a+b) = a 2 – b 2, and then the formula 1) ;
G) sin 2 αcosα – cosα. Let's take the common factor out of brackets.
sin 2 αcosα – cosα = cosα(sin 2 α – 1) = -cosα(1 – sin 2 α) = -cosα ∙ cos 2 α = -cos 3 α. You, of course, have already noticed that since 1 – sin 2 α = cos 2 α, then sin 2 α – 1 = -cos 2 α. In the same way, if 1 – cos 2 α = sin 2 α, then cos 2 α – 1 = -sin 2 α.
d) sin 2 α+1+cos 2 α = (sin 2 α+cos 2 α)+1 = 1+1 = 2;
e) sin 4 α+2sin 2 αcos 2 α+cos 4 α. We have: the square of the expression sin 2 α plus the double product of sin 2 α by cos 2 α and plus the square of the second expression cos 2 α. Let's apply the formula for the square of the sum of two expressions: a 2 +2ab+b 2 =(a+b) 2. Next we apply the formula 1) . We get: sin 4 α+2sin 2 αcos 2 α+cos 4 α = (sin 2 α+cos 2 α) 2 = 1 2 = 1;
and) tg 2 α – sin 2 αtg 2 α = tg 2 α(1 – sin 2 α) = tg 2 α ∙ cos 2 α = sin 2 α. Apply the formula 1) , and then the formula 2) .
Remember: tgα ∙ cosα = sinα.
Similarly, using the formula 3) available: ctgα ∙ sinα = cosα. Remember!
h) ctg 2 αcos 2 α – ctg 2 α = ctg 2 α(cos 2 α – 1) = ctg 2 α ∙ (-sin 2 α) = -cos 2 α.
And) cos 2 α+tg 2 αcos 2 α = cos 2 α(1+tg 2 α) = 1. We first took the common factor out of brackets, and simplified the contents of the brackets using the formula 7).
Convert expression:
We applied the formula 7) and obtained the product of the sum of two expressions by the incomplete square of the difference of these expressions - the formula for the sum of cubes of two expressions.
The article describes in detail the basic trigonometric identities. These equalities establish the relationship between sin, cos, t g, c t g of a given angle. If one function is known, another can be found through it.
Trigonometric identities to be considered in this article. Below we show an example of their derivation with an explanation.
sin 2 α + cos 2 α = 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 , 1 + c t g 2 α = 1 sin 2 α
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Let's talk about an important trigonometric identity, which is considered the basis of trigonometry.
sin 2 α + cos 2 α = 1
The given equalities t g 2 α + 1 = 1 cos 2 α, 1 + c t g 2 α = 1 sin 2 α are derived from the main one by dividing both parts by sin 2 α and cos 2 α. After which we obtain t g α = sin α cos α, c t g α = cos α sin α and t g α · c t g α = 1 - this is a consequence of the definitions of sine, cosine, tangent and cotangent.
The equality sin 2 α + cos 2 α = 1 is the main trigonometric identity. To prove it, you need to turn to the topic of the unit circle.
Let the coordinates of point A (1, 0) be given, which after rotation by an angle α becomes point A 1. By definition of sin and cos, point A 1 will receive coordinates (cos α, sin α). Since A 1 is located within the unit circle, this means that the coordinates must satisfy the condition x 2 + y 2 = 1 of this circle. The expression cos 2 α + sin 2 α = 1 should be valid. To do this, it is necessary to prove the main trigonometric identity for all rotation angles α.
In trigonometry, the expression sin 2 α + cos 2 α = 1 is used as the Pythagorean theorem in trigonometry. To do this, consider a detailed proof.
Using a unit circle, we rotate point A with coordinates (1, 0) around the central point O by angle α. After rotation, the point changes coordinates and becomes equal to A 1 (x, y). We lower the perpendicular line A 1 H to O x from point A 1.
The figure clearly shows that a right triangle O A 1 N has been formed. The modulus of the legs O A 1 N and O N are equal, the entry will take the following form: | A 1 H | = | y | , | O N | = | x | . The hypotenuse O A 1 has a value equal to the radius of the unit circle, | O A 1 | = 1 . Using this expression, we can write the equality using the Pythagorean theorem: | A 1 N | 2 + | O N | 2 = | O A 1 | 2. Let us write this equality as | y | 2 + | x | 2 = 1 2, which means y 2 + x 2 = 1.
Using the definition of sin α = y and cos α = x, we substitute the angle data instead of the coordinates of the points and move on to the inequality sin 2 α + cos 2 α = 1.
The basic connection between sin and cos of an angle is possible through this trigonometric identity. Thus, we can calculate the sin of an angle with a known cos and vice versa. To do this, it is necessary to resolve sin 2 α + cos 2 = 1 with respect to sin and cos, then we obtain expressions of the form sin α = ± 1 - cos 2 α and cos α = ± 1 - sin 2 α, respectively. The magnitude of the angle α determines the sign in front of the root of the expression. For a detailed explanation, you need to read the section on calculating sine, cosine, tangent and cotangent using trigonometric formulas.
Most often, the basic formula is used to transform or simplify trigonometric expressions. It is possible to replace the sum of the squares of sine and cosine by 1. Identity substitution can be either direct or reverse order: unit is replaced by the expression of the sum of the squares of sine and cosine.
Tangent and cotangent through sine and cosine
From the definition of cosine and sine, tangent and cotangent, it is clear that they are interconnected with each other, which allows you to separately convert the necessary quantities.
t g α = sin α cos α c t g α = cos α sin α
From the definition, sine is the ordinate of y, and cosine is the abscissa of x. Tangent is the relationship between the ordinate and abscissa. Thus we have:
t g α = y x = sin α cos α , and the cotangent expression has the opposite meaning, that is
c t g α = x y = cos α sin α .
It follows that the resulting identities t g α = sin α cos α and c t g α = cos α sin α are specified using sin and cos angles. The tangent is considered to be the ratio of the sine to the cosine of the angle between them, and the cotangent is the opposite.
Note that t g α = sin α cos α and c t g α = cos α sin α are true for any value of angle α, the values of which are included in the range. From the formula t g α = sin α cos α the value of the angle α is different from π 2 + π · z, and c t g α = cos α sin α takes the value of the angle α different from π · z, z takes the value of any integer.
Relationship between tangent and cotangent
There is a formula that shows the relationship between angles through tangent and cotangent. This trigonometric identity is important in trigonometry and is denoted as t g α · c t g α = 1. It makes sense for α with any value other than π 2 · z, otherwise the functions will not be defined.
The formula t g α · c t g α = 1 has its own peculiarities in the proof. From the definition we have that t g α = y x and c t g α = x y, hence we get t g α · c t g α = y x · x y = 1. Transforming the expression and substituting t g α = sin α cos α and c t g α = cos α sin α, we obtain t g α · c t g α = sin α cos α · cos α sin α = 1.
Then the expression of tangent and cotangent has the meaning of when we ultimately obtain mutually inverse numbers.
Tangent and cosine, cotangent and sine
Having transformed the main identities, we come to the conclusion that the tangent is related through the cosine, and the cotangent through the sine. This can be seen from the formulas t g 2 α + 1 = 1 cos 2 α, 1 + c t g 2 α = 1 sin 2 α.
The definition is as follows: the sum of the square of the tangent of an angle and 1 is equated to a fraction, where in the numerator we have 1, and in the denominator the square of the cosine of a given angle, and the sum of the square of the cotangent of the angle is the opposite. Thanks to the trigonometric identity sin 2 α + cos 2 α = 1, we can divide the corresponding sides by cos 2 α and get t g 2 α + 1 = 1 cos 2 α, where the value of cos 2 α should not be equal to zero. When dividing by sin 2 α, we obtain the identity 1 + c t g 2 α = 1 sin 2 α, where the value of sin 2 α should not be equal to zero.
From the above expressions we found that the identity t g 2 α + 1 = 1 cos 2 α is true for all values of the angle α not belonging to π 2 + π · z, and 1 + c t g 2 α = 1 sin 2 α for values of α not belonging to interval π · z.
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Basic trigonometric identities.
secα read: “secant alpha”. This is the reciprocal of cosine alpha.
cosecα read: “cosecant alpha.” This is the reciprocal of sine alpha.
Examples. Simplify the expression:
A) 1 – sin 2 α; b) cos 2 α – 1; V)(1 – cosα)(1+cosα); G) sin 2 αcosα – cosα; d) sin 2 α+1+cos 2 α;
e) sin 4 α+2sin 2 αcos 2 α+cos 4 α; and) tg 2 α – sin 2 αtg 2 α; h) ctg 2 αcos 2 α – ctg 2 α; And) cos 2 α+tg 2 αcos 2 α.
A) 1 – sin 2 α = cos 2 α according to the formula 1) ;
b) cos 2 α – 1 =- (1 – cos 2 α) = -sin 2 α also applied the formula 1) ;
V)(1 – cosα)(1+cosα) = 1 – cos 2 α = sin 2 α. First, we applied the formula for the difference of the squares of two expressions: (a – b)(a+b) = a 2 – b 2, and then the formula 1) ;
G) sin 2 αcosα – cosα. Let's take the common factor out of brackets.
sin 2 αcosα – cosα = cosα(sin 2 α – 1) = -cosα(1 – sin 2 α) = -cosα ∙ cos 2 α = -cos 3 α. You, of course, have already noticed that since 1 – sin 2 α = cos 2 α, then sin 2 α – 1 = -cos 2 α. In the same way, if 1 – cos 2 α = sin 2 α, then cos 2 α – 1 = -sin 2 α.
d) sin 2 α+1+cos 2 α = (sin 2 α+cos 2 α)+1 = 1+1 = 2;
e) sin 4 α+2sin 2 αcos 2 α+cos 4 α. We have: the square of the expression sin 2 α plus the double product of sin 2 α by cos 2 α and plus the square of the second expression cos 2 α. Let's apply the formula for the square of the sum of two expressions: a 2 +2ab+b 2 =(a+b) 2. Next we apply the formula 1) . We get: sin 4 α+2sin 2 αcos 2 α+cos 4 α = (sin 2 α+cos 2 α) 2 = 1 2 = 1;
and) tg 2 α – sin 2 αtg 2 α = tg 2 α(1 – sin 2 α) = tg 2 α ∙ cos 2 α = sin 2 α. Apply the formula 1) , and then the formula 2) .
Remember: tgα ∙ cosα = sinα.
Similarly, using the formula 3) available: ctgα ∙ sinα = cosα. Remember!
h) ctg 2 αcos 2 α – ctg 2 α = ctg 2 α(cos 2 α – 1) = ctg 2 α ∙ (-sin 2 α) = -cos 2 α.
And) cos 2 α+tg 2 αcos 2 α = cos 2 α(1+tg 2 α) = 1. We first took the common factor out of brackets, and simplified the contents of the brackets using the formula 7).
Convert expression:
The "sin" request is redirected here; see also other meanings. The "sec" request is redirected here; see also other meanings. The "Sine" request is redirected here; see also other meanings... Wikipedia
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The relationships between the basic trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this explains the abundance of trigonometric formulas. Some formulas connect trigonometric functions of the same angle, others - functions of a multiple angle, others - allow you to reduce the degree, fourth - express all functions through the tangent of a half angle, etc.
In this article we will list in order all the main trigonometric formulas, which are sufficient to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them by purpose and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities define the relationship between sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function in terms of any other.
For a detailed description of these trigonometry formulas, their derivation and examples of application, see the article.
Reduction formulas
Reduction formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, as well as the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.
The rationale for these formulas, a mnemonic rule for memorizing them and examples of their application can be studied in the article.
Addition formulas
Trigonometric addition formulas show how trigonometric functions of the sum or difference of two angles are expressed in terms of trigonometric functions of those angles. These formulas serve as the basis for deriving the following trigonometric formulas.
Formulas for double, triple, etc. angle
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. angle
Half angle formulas
Half angle formulas show how trigonometric functions of a half angle are expressed in terms of the cosine of a whole angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Degree reduction formulas
Trigonometric formulas for reducing degrees are designed to facilitate the transition from natural powers of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow you to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
The main purpose formulas for the sum and difference of trigonometric functions is to go to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow you to factor the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to a sum or difference is carried out using the formulas for the product of sines, cosines and sine by cosine.
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