Acquaintance with such a trigonometric function as sinus occurs in the school year of algebra. What does she represent? What properties do you have? How is the sinus with other functions of trigonometry, such as cosine, tangent and catangent?
Geometric definition of sinus
In order to formulate the definition of sinus, turn to a single circle. Its center will lie at the point of intersection of the X and Y axes of the Cartesian coordinate system. Denote this point as t. O, its coordinates - (0,0). Radius of this circle R \u003d 1. Next, we will build a rectangular triangle. For this:
- Take on a single circle an arbitrary T. P. Its coordinates - (x, y).
- After T. P, swipe the vertical that will form an angle of 90 ° with the Ox axis.
- The intersection point of this vertical with the OX axis will be denoted by T. L.
- As a result, the segments PL \u003d Y and OL \u003d X were formed.
- Connect T. P (x, y) and the beginning of the coordinate - t. O (0,0). Cut OP \u003d R \u003d 1.
- The resulting ∠lop is denoted as μ.
The sinus of the angle μ is called the ratio of the ordinate y (Pl) to the radius of the circle R (OP). Because The sections of PL and OP are respectively cathet and the hypothenoise of the triangle ΔOPL with ∠olp \u003d 90 °, then the concept of sine characterizes the ratio between the sides of the rectangular triangle.
The corner sinus is the ratio of the length of the opposite category to the length of the hypotenuse.
Definition of sinus for an arbitrary angle
Consider an arbitrary circle of radius B. ∠η formed by the axis of the abscisso o x. and Radius-vector OB (B x., B. y.) (T. B belongs to the circle). Power perpendicular from t. B on the axis of the abscissa and the axis of the ordinate. Based on the wording of the corner sinus for a rectangular triangle, it follows that
sinη \u003d B. y./ b.
The sinus of an arbitrary angle formed by the radius by the vector and the abscissa axis is the ratio of the projection of this vector on the ordinate axis to the length of the radius-vector.
Definition of sinus through trigonometric identities
Using the main identity of trigonometry (SINμ 2.+ COSμ. 2.\u003d 1), it is easy to notice that:
sinμ. 2.\u003d 1 - COSμ 2.⇒ ιsinμι \u003d √1 - COSμ 2
sinμ \u003d ± √1 - cosμ 2.
A positive or negative sinus value determines a quarter of the coordinate plane in which the angle falls. So, in the first and second quarters, the value of sinus will be positive. While in the third and fourth quarters, the function will take a negative value.
Sinus Function Chart and Properties
To build a graph of the Sinus function, move to the Cartesian coordinate system. Noting consistently values \u200b\u200bon the plane when moving along the axis o x., draw the schedule of the desired function. The following properties of sinus are clearly visible:
- The field definition area is all valid numbers.
- In this area, the value of the value is limited - from -1 to 1 inclusive.
- Function periodic. Repeat values \u200b\u200boccurs after 2π (i.e. 360 °)
- In this case, sin (- μ) \u003d - sinμ. So the sinus function is odd.
Definition of sinus through the formula
Returning to a single circle, you can see that:
sINμ \u003d Y / R. Because R \u003d 1, y / 1 \u003d y ⇒ sinμ \u003d y.
sin (π / 2 + η) \u003d cosη, sin (π + η) \u003d - sinη,
sin (π / 2 - η) \u003d cosη, sin (π - η) \u003d sinη,
sin (3π / 2 + η) \u003d -cosη, sin (2π + η) \u003d sinη,
sin (3π / 2 - η) \u003d -cosη, sin (2π - η) \u003d -sinη.
Because Sinusa has a function periodic and its period is 2π (360 °), the above relations are valid and generally:
sin (2πk + η) \u003d sinη,
sin (π / 2 + η + 2πk) \u003d cosηη, sin (π + η + 2πk) \u003d -sinη,
sin (π / 2 - η + 2πk) \u003d cosηη, sin (π - η + 2πk) \u003d sinη,
sin (3π / 2 + η + 2πk) \u003d -cosηη, sin (2π + η + 2πk) \u003d sinη,
sin (3π / 2 - η + 2πk) \u003d -cosηη, sin (2π - η + 2πk) \u003d -sinη, where k is any number from the range of valid numbers.
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