Difference between revisions of "Trigonometry"

Revision as of 01:21, 15 April 2015

Trigonometry seeks to find the lengths of a triangle's sides, given 2 angles and a side. Trigonometry is closely related to analytic geometry.

Basic definitions

Usually we call an angle $\theta$ , read "theta", but $\theta$ is just a variable. We could just as well call it $a$ .

For the following definitions, the "opposite side" is the side opposite of angle $\theta$ , and the "adjacent side" is the side that is part of angle $\theta$ , but is not the hypotenuse.

i.e. If ABC is a right triangle with right angle C, and angle A = $\theta$ , then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.

Sine

The sine of an angle $\theta$ , abbreviated $\sin \theta$ , is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\sin 30^{\circ}=\frac 12$ .

Cosine

The cosine of an angle $\theta$ , abbreviated $\cos \theta$ , is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\cos 30^{\circ} =\frac{\sqrt{3}}{2}$ .

Tangent

The tangent of an angle $\theta$ , abbreviated $\tan \theta$ , is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\tan 30^{\circ}=\frac{\sqrt{3}}{3}$ . (Note that $\tan \theta=\frac{\sin\theta}{\cos\theta}$ .)

Cosecant

The cosecant of an angle $\theta$ , abbreviated $\csc \theta$ , is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\csc 30^{\circ}=2$ . (Note that $\csc \theta=\frac{1}{\sin \theta}$ .)

Secant

The secant of an angle $\theta$ , abbreviated $\sec \theta$ , is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\sec 30^{\circ}=\frac{2\sqrt{3}}{3}$ . (Note that $\sec \theta=\frac{1}{\cos \theta}$ .)

Cotangent

The cotangent of an angle $\theta$ , abbreviated $\cot \theta$ , is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\cot 30^{\circ}=\sqrt{3}$ . (Note that $\cot \theta=\frac{\cos\theta}{\sin\theta}$ or $\cot \theta = \frac{1}{\tan \theta}$ .)

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 $\theta$ and the coordinates of the point we picked to be $(x,y)$ , we have:

$\begin{align*} \sin \theta &= y \\ \cos \theta &= x \\ \tan \theta &= \frac{y}{x} \\ \csc \theta &= \frac{1}{y} \\ \sec \theta &= \frac{1}{x} \\ \cot \theta &= \frac{x}{y} \end{align*}$

Note that $(x,y)$ is the rectangular coordinates for the point $(1,\theta)$ .

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.

Pythagorean Identities

$\sin^2\theta+\cos^2\theta=1$
$1+\tan^2\theta=\sec^2\theta$
$1+\cot^2\theta=\csc^2\theta$

Double-Angle Identities

$\sin 2\theta=2\sin\theta\cos\theta$
$\cos 2\theta=\cos^2\theta-\sin^2\theta$
$\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$

Half-Angle Identites

$\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$
$\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}$
$\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$

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