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,

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.

trigonometry [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle.

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 todayThere 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 geographyTrigonometric 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 physicsAlthough 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.

a

b

c

C

B

A

When labeling the parts of a triangle, use capital letters to name the angles and lower case letters to name the sides.

Notice that the side opposite the angle is named using the same letter (just lower case).

Adjacent means next to. There are two sides adjacent to A - side b and side c.

Which sides are adjacent to B? C?

a

b

c

C

B

A

A side that is opposite an angle is one that is across fromthe 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.

The sine of an angle is the ratio of the opposite side to the hypotenuse. The abbreviation for sine is sin.

a

b

c

C

B

A

hypotenuse

sin A = The length of the side opposite A

The length of the hypotenuse=

ac

sin B = bc

The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The abbreviation for cosine is cos.

a

b

c

C

B

A

hypotenuse

cos A = The length of the side adjacent to A

The length of the hypotenuse=

bc

cos B = ac

The tangent of an angle is the ratio of the opposite side to the adjacent side. The abbreviation for tangent is tan.

a

b

c

C

B

A

hypotenuse

tan A = The length of the side opposite A

The length of the side adjacent to A=

ab

tan B = ba

I get it!

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)

A

BC 73

7

14

Chief SohCahToa

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)

A

BC 180

19181

Chief SohCahToa

This is pretty easy!