Definition of the sine and cosine of an angle x
The definitions of the sine, cosine, tangent and cotangent functions can now be formulated for any circle or for the unit circle (i.e. a circle with radius r = 1 unit of length).
With respect to the unit circle - do my homework for money , the following definitions (equivalent to the definitions on any circle) apply:
The ordinate v of the point belonging to the angle x
P(u; v) on the unit circle is called sine of the angle x:
sin x= (average)PQ/(average)OP=v/1=v (x∈[0; 2π]).
The abscissa u of the point P belonging to the angle x on the unit circle is called cosine of the angle x:
cos x=(average)OQ/(average)OP=u/1=u (x∈[0; 2π])
The unique assignment x→sin x is called the sine function, and the unique assignment x→cos x is called the cosine function accordingly.
If we assume that the second leg of the angle x can be rotated around the origin as often as desired and, moreover, in the mathematically positive as well as in the mathematically negative sense of rotation - https://domyhomework.club/math-problem-solver/ , the concept of angle can be extended:
Each additional full rotation increases the angle by 2π or 360° (for counterclockwise rotation) or -2π or -360° (for clockwise rotation).
Thus, the entire set R of real numbers can be used as the domain of definition of the sine and cosine functions.
Thus, for example, it is valid:
The definition of the tangent of an angle x can be done immediately using the sine and cosine of this angle - microeconomics homework help , but also with reference to the unit circle and the tangent to it at the point (1; 0).
For any angle x (x∈R and x ≠ (2k+1)π/2, k∈Z), the quotient of the sine and cosine of this angle is called the tangent of the angle x.
The unique assignment x→ tan x is called the tangent function.
Correspondingly, the quotient of the cosine and the sine of an angle x (x∈R and x≠k π, k∈Z) is called the cotangent of x and the unique assignment x→ cot x is called the cotangent function.
Sine, cosine, tangent and cotangent functions are special angle functions or trigonometric functions.