Difference between revisions of "Trigonometry"
(→Cotangent) |
Flamefoxx99 (talk | contribs) m (→Trigonometery Definitions for non-acute angles) |
||
Line 32: | Line 32: | ||
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 <math>\theta </math> and the coordinates of the point we picked to be <math>(x,y) </math>, we have: | 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 <math>\theta </math> and the coordinates of the point we picked to be <math>(x,y) </math>, we have: | ||
− | < | + | <cmath>\begin{align*} \sin \theta &= y \ |
− | + | \cos \theta &= x \ | |
− | + | \tan \theta &= \frac{y}{x} \ | |
− | + | \csc \theta &= \frac{1}{y} \ | |
− | + | \sec \theta &= \frac{1}{x} \ | |
− | + | \cot \theta &= \frac{x}{y} \end{align*}</cmath> | |
Note that <math>(x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>. | Note that <math>(x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>. | ||
− | 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. | + | 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== | ==Trigonometric Identities== |
Revision as of 12:17, 19 October 2013
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
[hide]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 .)
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:
\begin{align*} \sin \theta &= y \\ \cos \theta &= x \\ \tan \theta &= \frac{y}{x} \\ \csc \theta &= \frac{1}{y} \\ \sec \theta &= \frac{1}{x} \\ \cot \theta &= \frac{x}{y} \end{align*} (Error compiling LaTeX. Unknown error_msg)
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.