Formulas for reducing trigonometric functions presentation. Presentation on the topic "reduction formulas"
This presentation is an excellent educational material on the topic “Reduction Formulas”. This is one of the important topics in the field of trigonometry which will be studied for a long time in class 10th.
The process will solve many algebraic and geometric problems using trigonometry terms.
The first slide of the presentation talks about the meaning of reduction formulas in trigonometry. Functions of a certain type can be simplified using these rules, which are the subject of this training material.
For certain signs of the function that will undergo transformations, the name of the trigonometric function is retained. In other cases, sines change to cosines, tangents to cotangents, and, accordingly, vice versa.
The next slide talks about how to place the sign correctly. These rules must be remembered.
All these reduction formulas can be written in terms of degrees. How this is done is shown on the next slide.
All these theoretically reviewed rules for reducing trigonometric functions are demonstrated in detail in a visual form below.
The numerical unit circle is shown with all the necessary notations, the periods are also visible, the arcs in question are indicated, and there is a table on which everything is demonstrated step by step with the help of animation effects.
There are 4 similar slides. All of them explain reduction formulas. After viewing all these slides, the student should understand the whole point.
The following is the first example. It suggests finding the sine of a certain degree, greater than 180. The sign is negative. Using the reduction formula solves this example much easier. Everything is also clearly demonstrated on the table.
The next slide contains a task in which you need to prove some identity. To prove it, another reduction formula is used.
The following examples are similar. On the right side of all statements there is a unit, which tells students what formula they should arrive at as a result.
The presentation will help you prepare for independent work that contains trigonometric expressions, to solve, prove or simplify which you need to understand the basic formulas, principles and methods.
Allows you to calculate the values of trigonometric angle functions any quarters through the corner I quarters
Municipal educational institution gymnasium No. 18 named after. V.G. Sokolova, Rybinsk
Pestova E.V. Mathematic teacher
For example: sin ( + α) = - sin α
cos (3 /2+ α) = sin α
sin ( + α) = - sin α cos (3 / 2 + α) = sin α
α – angle of the first quarter, i.e. α˂ / 2
II III IV I II III IV
sin ( + α) = - sin α cos (3 /2+ α) = sin α
cos ( - α) = - cos α sin ( /2+ α) = cos α
- How is the sign placed on the right side of an equality?
- In what case is the name of the original function replaced?
Rules:, if 0 ± α , 2 ± α name of the original function saved / 2 ± α , 3 / 2 ± α name of the original function replaced
For example: simplify cos ( - α) =
1 . - α – angle of the second quarter, cosine – negative, so we set “ minus ».
2. Angle - α is set aside from the OX axis, which means Name functions(cosine) saved .
Answer: cos ( - α) = - cos α
Rules: 1. The function on the right side of the equality is taken with the same sign as the original function, if 0 ± α , 2 ± α name of the original function saved. For angles that are laid off from the OU axis, / 2 ± α , 3 / 2 ± α name of the original function replaced(sine to cosine, cosine to sine, tangent to cotangent, cotangent to tangent).
For example: simplify sin (3 /2+ α) =
1 . 3 / 2 + α is the angle of the fourth quarter, the sine is negative, so we set “ minus ».
2. The angle 3 / 2 + α is set aside from the axis of the op-amp, which means function name(sinus) is changing to cosine.
Answer: sin (3 /2+ α) = - cos α
Simplify:
- sin ( + α) =
1). + α – angle... of a quarter, the sine in this quarter has the sign...
2). The angle + α is set aside from the axis ..., which means the name of the function (sine) ...
Answer: sin ( + α) = - sin α
- cos (3 /2+ α) =
1). Which quarter is the corner?
Answer: cos (3 /2+ α) = sin α
- sin (3 /2- α) =
1). Which quarter is the corner?
2). From which axis do we plot the angle? Should I change the name of the function?
Answer: sin (3 /2- α) = - cos α
- For calculations:
- To simplify expressions:
Prove these equalities in different ways
(using the rules learned and using the definition of tangent and cotangent).
On one's own. Simplify expressions:
- What new did you learn in the lesson?
- What have you learned?
- What rule do you remember?
- What are reduction formulas used for?
Slide 2
x y 0 cos sin 900+ 1800+ 2700+ Let’s construct an arbitrary acute angle of rotation . Now let's draw the angles 900+ , 1800+ , 2700+ and 3600+ . сos(900+) sin(900+) сos(1800+) sin(1800+) sin(2700+) cos(2700+) , 3600+ From the equality of right triangles we can conclude that: cos =sin(900+ )=–cos(1800+ )=–sin(2700+ )=cos(3600+ ), and also sin=–cos(900+ )=–sin(1800+ )=cos(2700+ )=sin(3600+ ).
Slide 3
The values of trigonometric functions of any angle of rotation can be reduced to the value of trigonometric functions of an acute angle. This is why reduction formulas are used. Let's try to understand the following table (transfer it to your notebook!): Everything is clear with the first column - it contains trigonometric functions that you know. The second column shows that any argument (angle) of these functions can be represented in this form. Let's explain this with specific examples:
Slide 4
In degrees: In radians: 10200=900·11+300=900·12–600 1020 90 11 90 120 90 30 As you can see, we used an action known to you from elementary school - division with a remainder. Moreover, the remainder does not exceed a divisor of 90 (in the case of a degree measure) or (in the case of a radian measure). Practice doing this! Multiply the resulting sum or difference by and get the required expressions. In any case, we have achieved the following: our argument to the trigonometric function is represented as an integer number of right angles plus or minus some acute angle. Let us now turn our attention to the 3rd and 4th columns of the table. Let us immediately note that in the case of an even number of right angles, the trigonometric function remains the same, and in the case of an odd number, it changes to a cofunction (sin to cos, tg to ctg and vice versa), and the argument of this function is the remainder.
Slide 5
It remains to deal with the sign in front of each result. These are the signs of these functions, depending on the coordinate quarters. Let us recall them: x 0 y 1 1 x 0 y 1 1 x 0 y 1 1 Signs sin Signs cos Signs tg and ctg + + + + + + – – – – – – Important! Do not forget to determine the sign of the final result using this function, and not the one obtained in the case of an even or odd number of right angles! Let's work on specific examples of how to use this table. Example 1. Find sin10200. Solution. First, let's present this angle in the form we need: 10200=900·11+300=900·12–600 I II
Slide 6
In the first case, we will have to change this sine function to a cofunction - cosine (the number of right angles is odd - 11), in the second the sine function will remain the same. I II The question of the sign of the result remains unclear. To solve it, we need to be able to work with the unit trigonometric circle (carefully watch the rotation of the point): ? ? x y 0 1 1 x y 0 1 1 I II 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 In any case, the fourth quarter is obtained, in which the sine is negative. – –
