Trigonometry

Trigonometry (literally the measure "metry" of triangles "trigons") is an important topic. It seems that no matter where you go, triangles are to be found.

If you want to learn trigonometry for the first time, try this: http://www.clarku.edu/~djoyce/trig/

Identities

Given a right triangle with sides of lengths A, B, and C, and with angles opposite those sides of a, b, and c, and where c is a right angle:

• C is called the hypotenuse. The hypotenuse is the side opposite the right angle.
• A and B are called the legs. The legs are the sides next to the right angle.

Pythagorean's Theorom

For fun, try to derive your own proofs. There are several thousand proofs out there.

The square of the length of the hypotenuse is the sum of the squares of the lengths of the legs in a right triangle.

<math>C^2 = A^2 + B^2 \,</math>

You can take the square root of both sides and get this common form:

<math>C = \sqrt{A^2 + B^2}</math>

Unit Circle

A unit circle has radius 1. Recall from Geometry that the circumference is <math>2 \pi r\,</math>.

Angles can be measured in radians. This is simply the length of the arc between the two legs of the angle on the unit circle. This is a more natural way to measure angles, given the function we will discuss below.

Degrees	Radians !
1°	π/180
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
360°	2π

You can build a right triangle to any point on the unit circle with legs that go horizontal and vertical. The length of these two legs is important in many problems, and so in trigonometry, we define some basic functions.

sin

sin (sine, pronounced "sign") is a function which takes an angle (usually, in radians), and gives a length between -1 and 1 (inclusive). This is the length of the vertical leg, if you measure angles starting to the right of the center.

Note that we rarely use parentheses for the trigonometric functions, since they are so common.

The following chart should give some common values of sin:

• <math>\sin 0 = \sin 0^\circ = 0 \,</math>
• <math>\sin a \approx a~\text{where}~a~\text{is very small}\,</math>
• <math>\sin \frac{\pi}{6} = \sin 30^\circ = \frac{1}{2} \,</math>
• <math>\sin \frac{\pi}{4} = \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \,</math>
• <math>\sin \frac{\pi}{3} = \sin 60^\circ = \frac{\sqrt{3}}{2} \,</math>
• <math>\sin \frac{\pi}{2} = \sin 90^\circ = 1 \,</math>

For all right triangles:

• <math>\sin \theta = \frac{opposite leg}{hypotenuse} \,</math>

Note the identities. sin is odd:

• <math>\sin -\theta = - \sin \theta \,</math>

sin rotates around a full circle:

• <math>\sin (\theta + 2\pi) = \sin \theta \,</math>
• <math>\sin (\theta + \pi) = -\sin \theta \,</math>

cosine

cos (cosine, pronounced 'CO-sign') give the other leg, the adjacent leg.

For all right triangles:

• <math>\cos \theta = \frac{adjacent leg}{hypotenuse} \,</math>

Sin and cos are related:

• <math>\cos (\theta + \frac{\pi}{2}) = \sin \theta \,</math>
• <math>\sin (\theta - \frac{\pi}{2}) = \cos \theta \,</math>

cos is even:

• <math>\cos - \theta = \cos \theta \,</math>

cos cycles:

• <math>\cos (\theta + \pi) = -\cos \theta \,</math>
• <math>\cos (\theta + 2\pi) = \cos \theta \,</math>

tangent

tan (tangent, pronounced "TAN-jent") gives the ratio of the legs.

For all right triangles:

• <math>\tan \theta = \frac{opposite leg}{adjacent leg} \,</math>

tan is not defined everywhere, particularly where <math>\cos \theta \,</math> would be 0. As it approaches these points, it tends towards positive or negative infinity, depending on the direction of the approach.

tan, like sin, is odd:

• <math>\tan - \theta = -\tan \theta \,</math>

And tan cycles:

• <math>\tan (\theta + \pi) = -\tan \theta \,</math>
• <math>\tan (\theta + 2\pi) = \tan \theta \,</math>

sec, csc, cot

Oftentimes, people work with fractions where the cos, sin, or tan are on the bottom.

• <math>\frac{1}{\cos \theta} = \sec \theta \,</math>, sec is secant, pronounced "SEE-cant".
• <math>\frac{1}{\sin \theta} = \csc \theta \,</math>, csc is cosecant, pronounced "CO-see-cant".
• <math>\frac{1}{\tan \theta} = \cot \theta \,</math>, cot is cotangent, pronounced "CO-tan-jent".

Remember that these functions are undefined where its partner is 0.

Inverse Functions

Sometimes you want to find an angle given a length. <math>n \,</math> is an integer, positive or negative. Typically, people prefer to keep the angles such that <math> 0 \le \theta < 2\pi \,</math>, or rarely, <math> -\pi < \theta \le \pi \,</math>.

• <math>\text{If}~\sin \theta = l~\text{then}~\arcsin l = \theta \pm 2n\pi\,</math>
• <math>\text{If}~\cos \theta = l~\text{then}~\arccos l = \theta \pm 2n\pi\,</math>
• <math>\text{If}~\tan \theta = l~\text{then}~\arctan l = \theta \pm 2n\pi\,</math>
• <math>\text{If}~\csc \theta = l~\text{then}~\arccsc l = \theta \pm 2n\pi\,</math>
• <math>\text{If}~\sec \theta = l~\text{then}~\arcsec l = \theta \pm 2n\pi\,</math>
• <math>\text{If}~\cot \theta = l~\text{then}~\arccot l = \theta \pm 2n\pi\,</math>

Of course, you have to keep in mind the problem of division by zero. Very large numbers may give very consistent answers for these functions.

Law of Sines and Cosines

For all triangles with sides of lengths A, B, and C and opposite angles a, b, and c:

• Law of Cosines: <math>C^2 = A^2 + B^2 - 2AB\cos c \,</math>
• Law of Sines: <math>\frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C} \,</math>

You can rearrange the Law of Cosines to find:

• The length of a third side given two adjacent sides and the angle between them.
• Any angle given three sides.

You can use the Law of Sines to find:

• The ratio of any side to the sin of its angle.
• Any angle given its opposite side and another side and its opposite angle.
• Any side given its opposite angle and another side and its opposite angle.

Using the above, given only a few facts about a triangle, you should be able to find everything about it.

Area of a Triangle

From Geometry, we learned that the area of a triangle is <math>A = \frac{1}{2}bh \,</math>.

Using Trigonometry, we can calculate the height given two sides and the angle between them: <math>A = \frac{1}{2} A B \cos c \,</math>.

Identities

A major component of Trigonometry is memorizing the identities. These come in rather handy. Here are all of the important ones you may ever run across.

<math>(\text{hypotenuse})^2 = (\text{leg a})^2 + (\text{leg b})^2 \, <math>\sin \theta = \frac{\text{opposite length}}{\text{hypotenuse length}} \,</math>

(In progress)