Trig

2 Pages 406 Words

Take an x-axis and an y-axis (orthonormal) and let o be the origin.
A circle centered in o and with radius = 1, is called a trigonometric circle or unit circle.
Turning counterclockwise is the positive orientation in trigonometry.
Angles are measured starting from the x-axis.
Two units to measure an angle are degrees and radians
An orthogonal angle = 90 degrees = pi/2 radians
In this theory we use mainly radians.
With each real number t corresponds just one angle, and just one point p on the unit circle, when we start measuring on the x-axis. We call that point the image point of t.





Examples:


with pi/6 corresponds the angle t and point p on the circle.
with -pi/2 corresponds the angle u and point q on the circle.
Trigonometric numbers of a real number t
With t radians corresponds 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)
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.





Analogous cotan(t) is the x-coordinate of the intersection point s' of the line op with the line y=1.



Basic formulas
With t radians corresponds exactly one point p(cos(t),sin(t)) on the unit circle. The square of the distance [op] = 1. Calculating this distance with the coordinates of p we have for each t :



cos²(t) + sin²(t) = 1

sin²(t) cos²(t)+sin²(t) 1...