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TRIGONOMETRY

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Nonetheless, one application of trigonometry is astronomy. As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy ... – PowerPoint PPT presentation

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Title: TRIGONOMETRY


1
TRIGONOMETRY
2
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.
3
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.
4
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.
5
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.
6
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.
7
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.
8
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.
9
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.
10
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?
11
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?
12
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.
13
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
14
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
15
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
16
How do I remember all of that?
17
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)
18
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)
19
Homework time!
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