Trigonometric circle and angles

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Choose an x-axis and a y-axis (orthonormal) and let O be the origin.
A circle of radius one centered at O is called 'the' trigonometric circle or 'the' unit circle.
Turning counterclockwise is the positive orientation in trigonometry.
Angles are measured starting from the x-axis.
The units used to measure an angle are 'degree' and 'radian'.
A right angle is an angle whose measure is exactly 90 degrees or pi/2 radians.
In this theory we use mainly radians.
Each real number t corresponds to exactly one angle, and to exactly one point P on the unit circle.
We call that point the 'image point' of t.
  • pi/6 corresponds to the angle t and to point P on the circle.
  • -pi/2 corresponds to the angle u and to point Q on the circle.

Trigonometric numbers of a real number t

The real number t corresponds to exactly one point P on the unit circle.
  • The x-coordinate of P is called the cosine of t. We write cos(t).
  • The y-coordinate of P is called the sine of t. We write sin(t).
  • The number sin(t)/cos(t) is called the tangent of t. We write tan(t).
  • The number cos(t)/sin(t) is called the cotangent of t. We write cot(t).
  • The number 1/cos(t) is called the secant of t. We write sec(t)
  • The number 1/sin(t) is called the cosecant of t. We write csc(t) or cosec(t)
The line with equation sin(t).x - cos(t).y = 0
contains the origin and point P(cos(t),sin(t)). So this line is OP.
On this line we take the intersection point S(1,?) with the line x = 1.
It is easy to see that ? = tan(t).
So tan(t) is the y-coordinate of the point S.

In an analogous manner we find that cotan(t) is the x-coordinate of the intersection point S' of the line OP with the line y = 1.

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