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TRIGONOMETRY

TRIGONOMETRY

Q What is trigonometry? A Trigonometry is the

study of how the sides and angles of a triangle

are related to each other. Q WHAT? That's

all? A Yes, that's all. It's all about

triangles, and you can't get much simpler than

that. Q You mean trigonometry isn't some big,

ugly monster that makes students turn green,

scream, and die? A No. It's just triangles.

Some historians say that trigonometry was

invented by Hipparchus, a Greek mathematician. He

also introduced the division of a circle into 360

degrees into Greece.

Hipparchus is considered the greatest

astronomical observer, and by some the greatest

astronomer of antiquity. He was the first Greek

to develop quantitative and accurate models for

the motion of the Sun and Moon. With his solar

and lunar theories and his numerical

trigonometry, he was probably the first to

develop a reliable method to predict solar

eclipses.

What is trigonometry again?

trigonometry Gr.,measurement of triangles, a

specialized area of geometry concerned with the

properties of and relations among the parts of a

triangle.

What can you do with trig?

Historically, it was developed for astronomy and

geography, but scientists have been using it for

centuries for other purposes, too. Besides other

fields of mathematics, trig is used in physics,

engineering, and chemistry. Within mathematics,

trig is used primarily in calculus (which is

perhaps its greatest application), linear

algebra, and statistics. Since these fields are

used throughout the natural and social sciences,

trig is a very useful subject to know.

Trigonometry today There are an enormous number

of applications of trigonometry. Of particular

value is the technique of triangulation which is

used in astronomy to measure the distance to

nearby stars, in geography to measure distances

between landmarks, and in satellite systems.

Other fields which make use of trigonometry

include astronomy (and hence navigation, on the

oceans, in aircraft, and in space), music theory,

acoustics, optics, analysis of financial markets,

electronics, probability theory, statistics,

biology, medical imaging (CAT scans and

ultrasound), pharmacy, chemistry, number theory

(and hence cryptology), seismology, meteorology,

oceanography, many physical sciences, land

surveying and geodesy, architecture, phonetics,

economics, electrical engineering, mechanical

engineering, civil engineering, computer

graphics, cartography, crystallography.

Click here to skip the application descriptions

and move straight to the basics.

Astronomy and geography Trigonometric tables were

created over two thousand years ago for

computations in astronomy. The stars were thought

to be fixed on a crystal sphere of great size,

and that model was perfect for practical

purposes. Only the planets (Mercury, Venus, Mars,

Jupiter, Saturn, the moon, and the sun) moved on

the sphere. The kind of trigonometry needed to

understand positions on a sphere is called

spherical trigonometry. Spherical trigonometry is

rarely taught now since its job has been taken

over by linear algebra. Nonetheless, one

application of trigonometry is astronomy.

As the earth is also a sphere, trigonometry is

used in geography and in navigation. Ptolemy

(100-178) used trigonometry in his Geography and

used trigonometric tables in his works. Columbus

carried a copy of Regiomontanus' Ephemerides

Astronomicae on his trips to the New World and

used it to his advantage.

Engineering and physics Although trigonometry was

first applied to spheres, it has had greater

application to planes. Surveyors have used

trigonometry for centuries. Engineers, both

military engineers and otherwise, have used

trigonometry nearly as long. Physics lays heavy

demands on trigonometry. All branches of physics

use trigonometry since trigonometry aids in

understanding space. Related fields such as

physical chemistry naturally use trig.

Some basics

When labeling the parts of a triangle, use

capital letters to name the angles and lower case

letters to name the sides.

B

Notice that the side opposite the angle is named

using the same letter (just lower case).

c

a

A

C

b

Adjacent means next to. There are two sides

adjacent to ?A - side b and side c.

Which sides are adjacent to ? B?

? C?

Some basics

B

c

a

A

C

b

A side that is opposite an angle is one that is

across from the angle. There is one side across

from ?A - side a.

Which side is opposite ? B?

? C?

Trigonometry is related to the acute angles in a

right triangle.

acute angles

There are three trigonometric ratios that relate

the measure of each acute angle to the lengths

of the sides in the triangle.

SINE

The sine of an angle is the ratio of the opposite

side to the hypotenuse. The abbreviation for

sine is sin.

B

c

hypotenuse

a

A

C

b

The length of the side opposite ?A

a c

sin A

The length of the hypotenuse

b c

sin B

COSINE

The cosine of an angle is the ratio of the

adjacent side to the hypotenuse. The

abbreviation for cosine is cos.

B

c

hypotenuse

a

A

C

b

The length of the side adjacent to ?A

b c

cos A

The length of the hypotenuse

a c

cos B

TANGENT

The tangent of an angle is the ratio of the

opposite side to the adjacent side. The

abbreviation for tangent is tan.

B

c

hypotenuse

a

A

C

b

The length of the side opposite ?A

a b

tan A

The length of the side adjacent to ? A

b a

tan B

How do I remember all of that?

Chief SohCahToa

A

I get it!

14

7

B

C

7?3

What is sin A? (leave your answer in fraction

form)

What is cos A? (leave your answer in fraction

form)

What is tan A? (leave your answer in fraction

form)

Chief SohCahToa

A

This is pretty easy!

181

19

B

C

180

What is sin B? (leave your answer in fraction

form)

What is cos B? (leave your answer in fraction

form)

What is tan B? (leave your answer in fraction

form)

Homework time!