Trigonometry

1
Trigonometry
A mathematics PowerPoint by Eric Zhao
2
Trigonometry is the study and solution of
Triangles. Solving a triangle means finding the
value of each of its sides and angles. The
following terminology and tactics will be
important in the solving of triangles.
Pythagorean Theorem (a2b2c2). Only for right
angle triangles Sine (sin), Cosecant (csc or
sin-1) Cosine (cos), Secant (sec or
cos-1) Tangent (tan), Cotangent (cot or
tan-1) Right/Oblique triangle
3
Part 1. Trigonometric Functions And Solving
Right Triangles
4
A trigonometric function is a ratio of certain
parts of a triangle. The names of these ratios
are The sine, cosine, tangent, cosecant, secant,
cotangent. Let us look at this triangle
Given the assigned letters to the sides and
angles, we can determine the following
trigonometric functions.
The Cosecant is the inversion of the sine, the
secant is the inversion of the cosine, the
cotangent is the inversion of the tangent.
With this, we can find the sine of the value of
angle A by dividing side a by side c. In order
to find the angle itself, we must take the sine
of the angle and invert it (in other words, find
the cosecant of the sine of the angle).
5
Try finding the angles of the following triangle
from the side lengths using the trigonometric
ratios from the previous slide.
Click for the Answer
The first step is to use the trigonometric
functions on angle A. Sin ? 6/10 Sin ?
0.6 Csc0.636.9 Angle A36.9 Because all angles
add up to 180, B90-11.53753.1
The measurements have changed. Find side BA and
side AC
Sin342/BA 0.5592/BA 0.559BA2 BA2/0.559 BA3.57
8
The Pythagorean theorem when used in this
triangle states that BC2AC2AB2 AC2AB2-BC2 AC2
12.802-48.802 AC8.8020.53
6
Part 2 Solving Oblique Triangles
7
When solving oblique triangles, simply using
trigonometric functions is not enough. You need
The Law of Sines
The Law of Cosines
a2b2c2-2bc cosA b2a2c2-2ac cosB c2a2b2-2ab
cosC
It is useful to memorize these laws. They can be
used to solve any triangle if enough measurements
are given.
8
REMEMBER
When solving a triangle, you must remember to
choose the correct law to solve it with. Whenever
possible, the law of sines should be used.
Remember that at least one angle measurement must
be given in order to use the law of sines. The
law of cosines in much more difficult and time
consuming method than the law of sines and is
harder to memorize. This law, however, is the
only way to solve a triangle in which all sides
but no angles are given. Only triangles with all
sides, an angle and two sides, or a side and two
angles given can be solved.
9
Solve this triangle
Click for answers
Because this triangle has an angle given, we can
use the law of sines to solve it. a/sin A b/sin
B c/sin C and subsitute 4/sin28º b/sin B
6/C. Because we know nothing about b/sin B, lets
start with 4/sin28º and use it to solve 6/sin C.
Cross-multiply those ratios 4sin C 6sin
28, divide 4 sin C (6sin28)/4. 6sin282.817.
Divide that by four 0.704. This means that
sin C0.704. Find the Csc of 0.704 º. Csc0.704º
44.749. Angle C is about 44.749º. Angle B is
about 180-44.749-2817.251. The last side is b.
a/sinA b/sinB, 4/sin28º b/sin17.251º,
4sin17.251sin28b, (4sin17.251)/sin28b.
b2.53.
10
Solve this triangle
Hint use the law of cosines
Start with the law of cosines because there are
no angles given. a2b2c2-2bc cosA. Substitute
values. 2.423.525.22-2(3.5)(5.2)
cosA, 5.76-12.25-27.04-2(3.5)(5.2) cos A,
33.5336.4cosA, 33.53/36.4cos A, 0.921cos A,
A67.07. Now for B. b2a2c2-2ac cosB,
(3.5)2(2.4)2(5.2)2-2(2.4)(5.2) cosB,
12.255.7627.04-24.96 cos B. 12.255.7627.04-24.
96 cos B, 12.25-5.76-27.04-24.96 cos B.
20.54/24.96cos B. 0.823cos B. B34.61. C180-34
.61-67.0778.32.
11
Part 3 Trigonometric Identities
12
Trigonometric identities are ratios and
relationships between certain trigonometric
functions. In the following few slides, you will
learn about different trigonometric identities
that take place in each trigonometric function.
13
Cofunctions
What is the sine of 60º? 0.866. What is the
cosine of 30º? 0.866. If you look at the name of
cosine, you can actually see that it is the
cofunction of the sine (co-sine). The cotangent
is the cofunction of the tangent (co-tangent),
and the cosecant is the cofunction of the secant
(co-secant). Sine60ºCosine30º Secant60ºCosecant3
0º tangent30ºcotangent60º
14
Other useful trigonometric identities
The following trigonometric identities are useful
to remember.
Sin ?1/csc ? Cos ?1/sec ? Tan ?1/cot ? Csc
?1/sin ? Sec ?1/cos ? Tan ?1/cot ?
(sin ?)2 (cos ?)21 1(tan ?)2(sec ?)2 1(cot
?)2(csc ?)2
15
Part 4 Degrees and Radians
16
Degrees and pi radians are two methods of showing
trigonometric info. To convert between them, use
the following equation. 2p radians 360
degrees 1p radians 180 degrees
Convert 500 degrees into radians. 2p radians
360 degrees, 1 degree 1p radians/180, 500
degrees p radians/180 500 500 degrees 25p
radians/9
17
Part 5 Review
18
Review 1
Write out the each of the trigonometric functions
(sin, cos, and tan) of the following degrees to
the hundredth place. (In degrees mode). Note
you do not have to do all of them ?
19
Review 2 a
Solve the following right triangles with the
dimensions given
20
Review 2 b
Solve the following oblique triangles with the
dimensions given
21
Review 3
Find each sine, cosecant, secant, and cotangent
using different trigonometric identities to the
hundredth place(dont just use a few identities,
try all of them.).
22
Review 4 a
Convert to radians
23
Review 4 b
Convert to degrees
24
Creator
Director
Producer
Author
Trigonometry
Eric Zhao
Eric Zhao
Eric Zhao
Eric Zhao
MathPower Nine, chapter 6
Basic Mathematics Second edition By Haym Kruglak,
John T. Moore, Ramon Mata-Toledo
Bibliography
The End
(at last)