Difference between revisions of "Trigonometry"
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− | The tangent of an angle <math>\theta</math>, abbreviated <math>\tan \theta</math>, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\tan 30^{\circ}=\frac{\sqrt{ | + | The tangent of an angle <math>\theta</math>, abbreviated <math>\tan \theta</math>, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\tan 30^{\circ}=\frac{\sqrt{1}}{3}</math>. (Note that <math> \tan \theta=\frac{\sin\theta}{\cos\theta}</math>.) |
===Cosecant=== | ===Cosecant=== |
Revision as of 11:49, 28 October 2019
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
or
.)
Trigonometry 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 or negative, or even 0.
Trigonometric Identities
There are many identities that are based on trigonometric functions.