How to find a sine?
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The study of geometry helps to develop thinking. This subject necessarily enters school preparation. In the life of the knowledge of this subject may be useful - for example, when planning an apartment.
From the history
As part of the geometry course, trigonometry is also studied, which explores trigonometric functions. In trigonometry, we study the sines, cosines, tangents, and cotangents of an angle.
But at the moment we start with the simplest - the sine. Let's consider in more detail the very first concept - the sine of an angle in geometry. What is a sine and how to find it?
The concept of "sine angle" and sinusoid
The sine of an angle is the ratio of the values of the opposite leg and the hypotenuse of a right triangle. This is a direct trigonometric function, which is denoted in writing as "sin (x)", where (x) is the angle of a triangle.
On the graph, the sine of an angle is indicated by a sine wave with its own characteristics. A sinusoid looks like a continuous wave-like line that lies within certain limits on the plane of coordinates.The function is odd; therefore, it is symmetric with respect to 0 on the coordinate plane (it comes out of the origin of coordinates).
The domain of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 pi. This means that every 2 Pi the pattern is repeated, and the sine wave passes a full cycle.
Sine wave equation
- sin x = a / c
- where a - opposite to the corner of the triangle leg
- c - hypotenuse of a right triangle
Properties of sine angle
- sin (x) = - sin (x). This feature demonstrates that the function is symmetric, and if the values of x and (–x) are set aside on both sides of the coordinate system, then the ordinates of these points will be opposite. They will be at equal distance from each other.
- Another feature of this function is that the graph of the function increases on the interval [- П / 2 + 2 Пn]; [П / 2 + 2Пn], where n is any integer. A decrease in the sine of the angle graph will be observed on the segment: [П / 2 + 2 Пn]; [3P / 2 + 2Pn].
- sin (x)> 0 when x lies in the range (2Pn, P + 2Pn)
- (x) <0 when x is in the range (-P + 2Pn, 2Pn)
The values of the angle sines are determined by special tables. Created such tables to facilitate the process of calculating complex formulas and equations.It is easy to use and contains values not only for the sin (x) function, but also for the values of other functions.
Moreover, the table of standard values of these functions is included in the compulsory study of memory, as the multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.
|angle value α (degrees)||0||15||30||45||60||75||90||120||135||150||180||270||360|
|the value of the angle α in radians (in terms of pi)||0||π/12||π/6||π/4||π/3||5π/12||π/2||2π/3||3π/4||5π/6||π||3π/2||2π|
|sin||0||√3-1 /2√2||1/2||√2/2||√3/2||√3+1 /2√2||1||√3/2||√2/2||1/2||0||-1||0|
There is also a table that defines the values of the trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of certain angles.
With trigonometric functions, equations are composed. It is easy to solve these equations if you know simple trigonometric identities and reductions of functions, for example, such as sin (П / 2 + х) = cos (x) and others. For such castings, a separate table was also compiled.
How to find the sine of the angle
When there is a task to find the sine of the angle, and by the condition we have only the cosine, tangent, or cotangent of the angle, we can easily calculate the necessary with the help of trigonometric identities.
- sin2x + cos2x = 1
Based on this equation, we can find both sine and cosine, depending on which value is unknown. We have a trigonometric equation with one unknown:
- sin2x = 1 - cos2x
- sin x = ± √ 1 - cos2x
- ctg2x + 1 = 1 / sin2x
From this equation, you can find the value of the sine, knowing the value of the cotangent of the angle. For simplicity, replace sin2x = y, and then you get a simple equation. For example, the value of cotangent is 1, then:
- 1 + 1 = 1 / y
- 2 = 1 / y
- 2u = 1
- y = 1/2
Now we perform the reverse replacement of the game:
- sin2x = ½
- sin x = 1 / √2
Since we took the cotangent value for the standard angle (450), the obtained values can be checked by.
If you have given the value of the tangent, and you need to find the sine, another trigonometric identity will help:
- tg x * ctg x = 1
It follows that:
- ctg x = 1 / tg x
In order to find the sine of a non-standard angle, for example, 2400It is necessary to use the formulas for reducing angles. We know that π with us corresponds to 1800. Thus, we express our equality using standard angles by decomposition.
- 2400= 1800+ 600
We need to find the following: sin (1800+ 600). In trigonometry, there are casting formulas that are useful in this case. This is the formula:
- sin (π + x) = - sin (x)
Thus, the sine of an angle of 240 degrees is:
- sin (1800+ 600) = - sin (600) = - √3/2
In our case, x = 60, and P, respectively, 180 degrees. The value (-√3 / 2) we found in the table of values of the functions of standard angles.
Thus you can expand non-standard angles, for example: 210 = 180 + 30.
In textbooks one can find many formulas for calculating trigonometric equations - subtraction, addition, product and division of trigonometric functions of different angles into each other, ascension to the power and transformation of one function into another using simple identities and many other operations.
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