Transcript 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.
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 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.
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
C
b
A
Adjacent means next to. There are two sides adjacent to
A - side b and side c.
Which sides are adjacent to  B?
 C?
B
c
a
C
A
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.
The sine of an angle is the ratio of the opposite side to the
hypotenuse. The abbreviation for sine is sin.
B
c
a
C
sin A =
hypotenuse
b
A
The length of the side opposite A
The length of the hypotenuse
sin B =
b
c
=
a
c
The cosine of an angle is the ratio of the adjacent side to the
hypotenuse. The abbreviation for cosine is cos.
B
c
a
C
cos A =
hypotenuse
A
b
The length of the side adjacent to A
The length of the hypotenuse
cos B =
a
c
b
= c
The tangent of an angle is the ratio of the opposite side to the
adjacent side. The abbreviation for tangent is tan.
B
c
a
C
tan A =
hypotenuse
A
b
The length of the side opposite A
The length of the side adjacent to  A
tan B =
b
a
=
a
b
Chief SohCahToa
A
I get it!
14
7
C
B
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
C
180
B
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)