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12
Introduction to Trigonometry
Concepts:
• Angles
– Initial Side and Terminal Side
– Standard Position
– Coterminal Angles
• Measuring Angles
– Radian Measure vs. Degree Measure
– Radian Measure as a Distance on the Unit Circle
– Converting between Radian Measure and Degree Measure
– Finding the Quadrant Associated with the Terminal Side of an Angle
• The
π
4
−
π
4
−
π
2
or the 45◦ − 45◦ − 90◦ Triangle
• The
π
6
−
π
3
−
π
2
or the 30◦ − 60◦ − 90◦ Triangle
• Identifying the Point on the Unit Circle that Corresponds to an Angle in Standard
Position
• The Trigonometric Functions
– The Definitions of sin, cos, tan, csc, sec, and cot Based on the Unit Circle
– Evaluating the Six Trigonometric Functions at Special Angles
– The Sign of a Trigonometric Function
• Approximating Values of Trigonometric Functions with Your Calculator
– Parentheses Are Important
– Radian Mode vs. Degree Mode
• Understanding Trigonometric Notation
• Trigonometric Identities
– Pythagorean Identities
– Periodicity Identities
– Negative Angle Identitites
(Sections 6.1-6.3)
1
12.1
Angles and Definitions
An angle is constructed by rotating a ray around its endpoint. This ray is called the initial
side of the angle. The ray obtained from the rotation is called the terminal side of the
angle. The direction of the rotation is indicated by an arrow.
Definition 12.1
An angle consists of
• A ray called the initial side of the angle.
• A ray called the terminal side of the angle. The initial side and the terminal side
must have the same endpoint.
• A direction and amount of rotation such that the terminal side of the angle
is obtained by rotating the initial side side of the angle around its endpoint in the
direction of rotation by the amount specified.
We can see that two different angles can have the same initial and terminal sides. When
this happens the angles are called coterminal angles.
Angles can be placed on a Cartesian coordinate system. When an angle is placed in the
Cartesian plane in such a way that the vertex of the angle is on the origin and the initial
side is on the positive x-axis, the angle is said to be in standard position.
12.2
Measuring Angles
The measure of an angle describes both the amount of rotation and the direction of the
rotation. We will deal with the amount of rotation momentarily. The sign of the measure
indicates the direction of the angle. When the rotation is in the counterclockwise direction,
the measure of the angle is positive. When it is in the clockwise direction, the measure of
the angle is negative.
Now we consider the amount of rotation. Undoubtedly, you associate “degrees” with angle
measurements. Degrees can be used to measure the amount of rotation, but these units are
rather arbitrary and often are not as useful as the units we will discuss today. (Think about
feet and meters when measuring length. You can use feet to measure length, but there are
2
advantages to using the metric system in the sciences because it is a base 10 system.)
There is another unit for angle measure called “radians.” Radian measure is particularly
useful because it is associated with a length. Place an angle in standard position on the
Cartesian coordinate system. Then place the unit circle on the same Cartesian coordinate
system. The unit circle is the circle with center at the origin and radius 1. It is the graph
of the equation x2 + y 2 = 1. Notice that (1, 0) is the intersection point of the initial side
of the angle and the unit circle. Follow this point around the unit circle as you rotate the
initial side of the angle around to the terminal side of the angle. The distance traveled
by the point (1, 0) around the unit circle along with the sign of the angle is the
radian measure of the angle. This has an advantage over degree measure because radian
measure is about both a distance and the amount of turn in the angle. Degree measure only
defines the amount of turn in the angle. Degree measure does not correspond to a distance.
y
2
1
0
-2
-1
0
1
2
−4π
radians
3
-1
distance=
−4π
units
3
-2
Example 12.2
Find the radian measure of a 360◦ angle.
Example 12.3
Find the radian measure of a 90◦ angle.
3
x
Example 12.4
Find the radian measure of a −60◦ angle.
Example 12.5
Find the radian measure of a 730◦ angle.
Example 12.6
Find the degree measure of an angle measuring
3π
radians.
4
Example 12.7
13π
radians in standard position. In which quadrant does the
Sketch an angle of measure
3
terminal side of the angle lie?
Example 12.8
−27π
Sketch an angle of measure
radians in standard position. In which quadrant does the
4
terminal side of the angle lie?
Example 12.9
In which quadrant does the terminal side of the angle which measures 1 radian lie?
4
12.3
Special Triangles
There are two special triangle that will be helpful when we are studying trigonometry.
The
π
4
−
π
4
−
or the 45◦ − 45◦ − 90◦ Triangle:
π
2
π
4
1
π
4
The
π
6
−
π
3
−
or the 30◦ − 60◦ − 90◦ Triangle
π
2
1
π
3
π
6
Example 12.10
Suppose than an angle of measure t radians intersects the unit circle at the point
What is one possibility for t? Is there more than one correct answer?
5
√ !
1
3
,−
.
2
2
12.4
The Trigonometric Functions
Whenever we discuss angles in this class, we will assume that the angle is in standard position
unless otherwise noted. This will allow us to discuss the point where the terminal side of
the angle intersects the unit circle. (The initial side will always intersect the unit circle at
.) For now, let’s denote the intersection point of the terminal side and the
unit circle by (a, b).
y
2
1
0
-2
-1
0
-1
1
2
x
(a, b)
-2
We will be very interested in the exact values of a and b for certain angles.
Example 12.11
Find P if the angle has measure
3π
.
2
Example 12.12
Find P if the angle has measure 5π.
Example 12.13
Find P if the angle has measure
4π
.
3
6
Have you heard of the functions sin, cos, and tan before? What words come to mind when
you think about these functions and their definitions? Can you apply your definitions to
find sin(100o )?
There are six basic trigonometric functions. Their names are sin (said “sine”), cos (said
“cosine”), tan (said “tangent”), csc (said “cosecant”), sec (said “secant”), and cot (said
“cot”). The point P is used in the definition of all six of these function.
y
2
1
0
-2
-1
0
-1
1
2
x
(a, b)
-2
Definition 12.14 (The Trigonometric Functions)
Let t be a real number. Place an angle of measure t in standard position and let P (a, b) be
the intersection point of the terminal side of the angle and the unit circle. Then
• sin(t) = b
• cos(t) = a
• tan(t) =
b
a
• csc(t) =
1
b
1
a
a
• cot(t) =
b
• sec(t) =
Note 12.15
The coordinates of P can be written as (
,
7
).
Example 12.16
Evaluate the six trigonometric functions at t = π.
Example 12.17
Evaluate the six trigonometric functions at t =
Example 12.18
Evaluate the six trigonometric functions at t =
8
7π
.
6
−29π
.
4
Example 12.19
An angle, θ, is placed in standard position. The terminal side of the angle passes through
the point (3, −4). Evaluate sin(θ), cos(θ), tan(θ), csc(θ), sec(θ), and cot(θ).
Example 12.20
The terminal side of an angle, x, in standard position contains the point (10, 3). Evaluate
the six trigonometric functions at x.
Example 12.21
Find all the solutions of the equation.
tan(x) = 1
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Example 12.22 (A Challenge)
This is number 77 from section 6.2 of your textbook.
Using only the definition and no calculator, determine which number is larger: sin(cos(0))
or cos(sin(0)).
Example 12.23
The terminal ray of an angle in standard position must be in one of the four quadrants or
on an axis. The sign of the trigonometric functions is determined by the position of the
terminal ray of the angle. Discuss the signs of the trigonometric functions and the position
of the terminal ray.
10
12.5
Trigonometric Functions and Your Calculator
Your Calculator can be used to approximate the values of trigonometric functions.
Example 12.24
Use your calculator to approximate sin(π).
As we discussed before, all angle measures in this class are in radians unless otherwise stated.
Therefore it is important for you to know how to switch between degree mode and radian
mode on your calculator. If you have a TI-83, hit MODE and choose degree. Now calculate
sin(π). Is this different?
Example 12.25
π Use your calculator to approximate cot
.
7
Make sure that your calculator is in radian mode for this class.
Example 12.26
Approximate each of the following on your calculator.
• sin(3 − 1)
• sin(3) − 1
• sin(3 − 1 (This is not a typo.)
What does this example illustrate?
Example 12.27
Approximate each of the following on your calculator.
• cos(32 )
• cos(3)2
• (cos(3))2
What does this example illustrate?
11
12.6
Trigonometric Notation
In this class, we have always used parentheses around the arguments of the trigonometric
functions. I still think that this is the best practice, but the fact of the matter is that
sometimes people do not include these. We need to understand what is meant by various
common notations.
Example 12.28
• What is meant by sin3 (t)?
• What is meant by sin(t3 )?
• What is meant by sin t3 ?
Example 12.29
If you want to approximate sin3 (.5) using your calculator, what should you type?
Example 12.30
Which of the following can be simplified?
•
sin(t)
cos(t)
sin2 (t)
•
t
•
sin(2t)
2
sin2 (t)
•
sin(t)
•
sin(t2 )
sin(t)
12
12.7
Some Identities
12.7.1
The Pythagorean Identities
Theorem 12.31 (The Pythagorean Identity for sin and cos)
For every real number t,
sin2 (t) + cos2 (t) = 1
.
PROOF:
Example 12.32
If π2 < t < π, and cos(t) =
−4
,
7
find exact values for the other trigonometric functions at t.
Example 12.33
Find two other Pythagorean Identities that involve the other trigonometric functions.
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12.7.2
Periodicity Identities
Think about the unit circle as you discover the periodicity identities. It might also help to
think of tan(θ) as the slope of the terminal side of the angle θ. Why?
Theorem 12.34
• sin(x ±
) = sin(x)
• cos(x ±
) = cos(x)
• tan(x ±
) = tan(x)
• csc(x ±
) = csc(x)
• sec(x ±
) = sec(x)
• cot(x ±
) = cot(x)
12.7.3
Negative Angle Identities
Think about the unit circle as you discover the periodicity identities.
Theorem 12.35 (Negative Angle Identities)
• sin(−x) =
• cos(−x) =
• tan(−x) =
• csc(−x) =
• sec(−x) =
• cot(−x) =
14