Trigonometry
Trigonometry seeks to find the lengths of a triangle's sides, given 2 angles and a side. Trigonometry is closely related to analytic geometry.
Contents
Basic definitions
Usually we call an angle , read "theta", but is just a variable. We could just as well call it .
For the following definitions, the "opposite side" is the side opposite of angle and the "adjacent side" is the side that is part of angle but is not the hypotenuse.
i.e. If ABC is a right triangle with right angle C, and angle A = , then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.
Sine
The sine of an angle , abbreviated , is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .
Cosine
The cosine of an angle , abbreviated , is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .
Tangent
The tangent of an angle , abbreviated , is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Cosecant
The cosecant of an angle , abbreviated , is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Secant
The secant of an angle , abbreviated , is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Cotangent
The cotangent of an angle , abbreviated , is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Trigonometery Definitions for non-acute angles
Consider a unit circle that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a hypotenuse 1 unit long. Letting the angle at the origin be and the coordinates of the point we picked to be we have:
Note that is the rectangular coordinates for the point
This is true for all angles (Even negative angles and angles greater than 360 degrees.) Due to the way trig ratios are defined for non acute angles, the value of a trig ratio could be positive of negative or even 0.