This is the last and most important lesson needed to solve problems B11. We already know how to convert angles from a radian measure to a degree measure (see the lesson “Radian and degree measure of an angle”), and we also know how to determine the sign of a trigonometric function, focusing on the coordinate quarters (see the lesson “Signs of trigonometric functions”).
The only thing left to do is calculate the value of the function itself - the very number that is written in the answer. This is where the basic trigonometric identity comes to the rescue.
Basic trigonometric identity. For any angle α the following 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:
Note 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 there were any).
That is why in all problems B11, which are found in the Unified State Examination 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 probably ask: “What about tangent and cotangent?” It is impossible to directly calculate these functions from the above formulas. However, there are important consequences from the basic trigonometric identity, which 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 derived from the main identity - it is enough to divide both sides by cos 2 α (to obtain the tangent) or by sin 2 α (to obtain the cotangent).
Let's look at all this with specific examples. Below are the real B11 problems, which are taken from the trial versions of the Unified State Examination in Mathematics 2012.
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°).
Consequently, angle α lies in the II coordinate quarter - all 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. Let's 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, angle α belongs to the interval (π 3π /2). Let's convert the angles from radian measures to degrees - we get: α ∈ (180°; 270°).
Obviously, this is the III coordinate quarter, where all cosines are negative. Therefore cos α = −0.5.
Task. Find tan α if the following is known:
Tangent and cosine are related by the equation following from the basic trigonometric identity:
We get: tan 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 radian measures to degrees - we get α ∈ (270°; 360°).
Obviously, this is the IV coordinate quarter, where all tangents are negative. Therefore tan α = −3.
Task. Find cos α if the following is known:
Again the sine is known and the cosine is unknown. Let us 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 there are positive. Therefore, cos α = 0.6.
Task. Find sin α if the following is known:
Let us write down a formula that follows from the basic trigonometric identity and directly connects 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 angle α ∈ (0; π /2). In degree measure, this is written as follows: α ∈ (0°; 90°) - I coordinate quarter.
So, the angle is in the I coordinate quadrant - all trigonometric functions there are positive, so sin α = 0.2.
Trigonometric functions- 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
Tan
Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are a type of elementary functions. Typically these include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Cosine- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are a type of elementary functions. Typically these include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Cotangent- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are a type of elementary functions. Typically these include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
Secant- Rice. 1 Graphs of trigonometric functions: sine, cosine, tangent, secant, cosecant, cotangent Trigonometric functions are a type of elementary functions. Typically these include sine (sin x), cosine (cos x), tangent (tg x), cotangent (ctg x), ... ... Wikipedia
History of trigonometry- Geodetic measurements (XVII century) ... Wikipedia
Tangent of half angle formula- In trigonometry, the tan of a half angle formula relates the tangent of a half angle to the trigonometric functions of a full angle: Variations of this formula are as follows... Wikipedia
Trigonometry- (from the Greek τρίγονο (triangle) and the Greek μετρειν (measure), that is, measurement of triangles) a branch of mathematics in which trigonometric functions and their applications to geometry are studied. This term first appeared in 1595 as... ... Wikipedia
Solving triangles- (lat. solutio triangulorum) a historical term meaning the solution of the main trigonometric problem: using known data about a triangle (sides, angles, etc.) find its remaining characteristics. The triangle can be located on... ... Wikipedia
Books
- Set of tables. Algebra and the beginnings of analysis. Grade 10. 17 tables + methodology, . The tables are printed on thick printed cardboard measuring 680 x 980 mm. The kit includes a brochure with teaching guidelines for teachers. Educational album of 17 sheets... Buy for 3944 RUR
- Tables of integrals and other mathematical formulas, Dwight G.B.. The tenth edition of the famous reference book contains very detailed tables of indefinite and definite integrals, as well as a large number of other mathematical formulas: series expansions, ...
In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish a connection between the sine, cosine, tangent and cotangent of one angle, and allow one to find any of these trigonometric functions through a known other.
Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.
Page navigation.
Relationship between sine and cosine of one angle
Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind 
. The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and, respectively, and the equalities 
And 
follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.
That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.
Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.
The basic trigonometric identity is very often used when converting trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in the reverse order: unit is replaced by the sum of the squares of the sine and cosine of any angle.
Tangent and cotangent through sine and cosine
Identities connecting tangent and cotangent with sine and cosine of one angle of view and 
follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, 
, and the cotangent is the ratio of the abscissa to the ordinate, that is, 
.
Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.
In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .
Relationship between tangent and cotangent
An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out a little differently. Since , That 
.
So, the tangent and cotangent of the same angle at which they make sense are .
To use presentation previews, create a Google account and log in to it: https://accounts.google.com
Slide captions:
Even if English is dear to someone, Chemistry is important to someone, Without mathematics for all of us But neither here nor there Equations are like poems for us And sines support our spirit Cosines are like songs for us, And trigonometry formulas Caress our ears!
Lesson topic: “Basic trigonometric identities. Problem solving.” Know: Be able to: Objective of the lesson:
I KNOW! I CAN! I WILL DECIDE! I
What is the unit circle called? x y α R
What directions of rotation of a unit radius are known? x y α R
In what units is the angle of rotation of a unit radius measured? x y α R
What is an angle of one radian? Approximately how many degrees does an angle of 1 radian contain? x y α R
Formulate the rules for converting from a degree measure of an angle to a radian measure and vice versa.
Formulate the rules for converting from a degree measure of an angle to a radian measure and vice versa. 30 0 π 45 0 π 2 2 π
What trigonometric functions do you know?
What trigonometric functions do you know? What determines the meaning of trigonometric functions?
Which quarter angle is the angle α if: α =15° α =190° α =100°
Which quarter angle is the angle α if: α =-20° α =-110° α =289°
Working in groups Rules for working in a group: The group discusses and decides together, puts forward ideas or refutes them. Each group member must work to the best of his ability. While working, treat your colleagues with respect: accepting or rejecting an idea, do it politely. Remember that everyone has the right to make mistakes. Remember that the success of the group depends on how well everyone demonstrates their strengths.
Group work
0° 30° 45° 60° 90° sin cos tg ctg 0 1 1 0 0 1 - - 1 0 Table of trigonometric function values
1 A 2 B 3 C 4 D 5 E 6 H 7 through K 8 L 9 through and M 10 through and N 1 - cos 2 α 1-sin 2 α sin 2 α Evaluation criteria: 10 tasks - grade “5”. 8-9 tasks – score “4”. 5-7 tasks – score “3”. 1-4 tasks – score “2”. Establish a correspondence between the left and right sides of the identity.
1 M 2 L 3 N 4 E 5 B 6 C 7 through A 8 K 9 through and H 10 through and D 1 - cos 2 α 1-sin 2 α sin 2 α Evaluation criteria: 10 tasks - grade “5”. 8-9 tasks – score “4”. 5-7 tasks – score “3”. 1-4 tasks – score “2”. Establish a correspondence between the left and right sides of the identity.
Basic trigonometric identity "trigonometric unit"
Basic trigonometric identity “trigonometric unit” Cosine square Very glad. Brother Sine Square is coming to see him! When they meet, the circle will be surprised: A whole family will come out, That is, a unit!
1. 3 sin 2 α + 3 cos 2 α 2. (1 – cos α)(1 + cos α) at α =90° 3. 1- sin 2 40 0 4. 5. tg α∙ ctg α 6. ( ctg 2 α + 1)(1 – sin 2 α) 7. tg α∙ ctg α -1 8. cos 2 α + ctg 2 α + sin 2 α and s t P to 1 cos 2 40° 3 ctg 2 α 0 1 2 3 4 5 6 7 8 Get the name of the mathematician in whose book the term “trigonometry” first appears. 1 2 3 4 5 6 7 8 P i t i c k u s 2-2 cos(-60 0)
Pitiscus
Al-Batuni Al-Khwarizmi
Bhaskara Nasireddin Tusi
Leonard Euler
Given the value of the trigonometric function, find the value of another function Quarter Given: Find: Solution: I sinα= 0.6 II cosα= sinα III tgα= ctgα IV cosα= tgα
Given the value of the trigonometric function, find the value of another function Quarter Given: Find: Solution: I sinα= 0.6
Given the value of the trigonometric function, find the value of another function Quarter Given: Find: Solution: II cosα= sinα = =
Given the value of the trigonometric function, find the value of another function Quarter Given: Find: Solution: III tgα= ctgα ctgα = = =
Given the value of the trigonometric function, find the value of another function Quarter Given: Find: Solution: IV cosα = tgα tgα = = = = = =
Application of trigonometry in human life.
Homework Message: “Trigonometry in human life” No. 304 p. 111
y=sinx Thanks for the lesson!
1 sin 240° 8 cos 290° 2 tg 98° 9 tg(-120°) 3 sin 70° 10 sin 4 ctg 200° 11 cos 5 cos 113° 12 cos 6 sin (- 140°) 13 sin 7 cos (- 300 °) 14 tg Determine the sign of the expression - - - - - - + + + + + + + +
On the topic: methodological developments, presentations and notes
The presentation presents solutions to key problems of a school mathematics course on finding all types of distances and angles in space using an algorithm, which allows it to be used both in studying...
Presentation for the lesson: "Angle between planes. Solving the problem using various methods"
This presentation can be used for clarity in revision lessons, to prepare for the Unified State Exam when solving problems of type C-2....