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Math 1330 - Section 4.1
Special Triangles
In this section, we’ll work with some special triangles before moving on to defining the six
trigonometric functions.
Two special triangles are 30 − 60 − 90 triangles and 45 − 45 − 90 triangles. With little
additional information, you should be able to find the lengths of all sides of one of these special
triangles.
First we’ll review some conventions when working with triangles. We label angles with capital
letters and sides with lower case letters. We’ll use the same letter to refer to an angle and the
side opposite it, although the angle will be a capital letter and the side will be lower case. In this
section, we will work with right triangles. Right triangles have one angle which measures 90
degrees, and two other angles whose sum is 90 degrees. The side opposite the right angle is
called the hypotenuse, and the other two sides are called the legs of the triangle. A labeled right
triangle is shown below.
B
c
a
A
b
C
Recall: When you are working with a right triangle, you can use the Pythagorean Theorem to
help you find the length of an unknown side. In a right triangle ABC with right angle C,
a2 + b2 = c2 .
Example: In right triangle ABC with right angle C, if AB = 12 and AC = 8, find BC.
1
Special Triangles
In a 30 − 60 − 90 triangle, the lengths of the sides are proportional. If the shorter leg (the side
opposite the 30 degree angle) has length a, then the longer leg has length 3a and the
hypotenuse has length 2a. An easy way to remember this is to write the lengths in ratio form as
a : 3a : 2a. So if you know (or can find) a, you can find the lengths of all three sides.
Example 1: Find x.
60 °
x
60 °
10
5
Example 2: Find x.
x
Example 3: Find x.
x
30°
4 2
2
In a 45 − 45 − 90 triangle, the two legs of the triangle are equal in length. This triangle is
also called an isosceles right triangle. If the legs each measure a, then the hypotenuse has length
2a . The ratio to remember for these triangles is a : a : 2a. Again, if you know a, you can
find the lengths of all three sides.
Example 4: Find x.
x
12
45 °
Example 5: Find x.
16
x
Example 6: In right triangle ABC with right angle C, AC = 4 and m(B) = 30° , find AB.
3
The Six Trigonometric Functions of an Angle
B
c
a
A
b
C
Suppose we are given a right triangle, ABC where ∠C = 90°. We define the trigonometric
functions of either of the acute angles of the triangle as follows:
Sine Function:
sin θ =
length of the side opposite θ
length of hypoteneuse
Cosine Function:
cos θ =
length of the side adjacent to θ
length of hypoteneuse
Tangent Function:
tan θ =
length of the side opposite θ
length of the side adjacent to θ
Cotangent Function:
cot θ =
length of the side adjacent to θ
length of side opposite to θ
Secant Function:
sec θ =
length of the hypoteneuse
length of the side adjacent to θ
Cosecant Function:
csc θ =
length of the hypoteneuse
length of the side opposite θ
Note: For acute angles the values of the trigonometric functions are always positive since they
are ratios of lengths.
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opposite
a
=
hypotenuse h
adjacent
b
=
cos( A) =
hypotenuse h
opposite a
tan( A) =
=
adjacent b
adjacent b
cot( A) =
=
opposite a
hypotenuse h
=
sec( A) =
adjacent
b
hypotenuse h
=
csc( A) =
opposite
a
sin( A) =
5
Example 1: Suppose you are given this triangle.
B
10
7
A
b
C
a. Find b.
b. Find the six trigonometric functions of angle A.
c. Find the six trigonometric functions of angle B.
6
Example 2: Suppose that BAC is a right triangle with A = 90° . If BC = 9 and AB = 6, find
each of the following:
sin(C ) =
tan(C ) =
cos( B ) =
tan( B ) =
Example 3: Suppose that θ is an acute angle in a right triangle and sec θ =
cos θ and tan θ .
5 3
. Find the
4
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(Extra) Example 4: Suppose that ∆DEF is a right triangle and D is an acute angle. If
4
sin D = , find the other five trigonometric functions of angle D.
5
Cofunctions
A special relationship exists between sine and cosine, tangent and cotangent, secant and
cosecant. The relationship can be summarized as follows:
Function of θ = Cofunction of the complement of θ
Recall that two angles are complementary if their sum is 90° . In example 10, angles B and C
are complementary, since A = 90° . Notice that sin C = cos B and sin B = cos C . The same will
be true of any pair of cofunctions.
Cofunction relationships
sin A = cos(90° − A)
cos A = sin(90° − A)
tan A = cot(90° − A)
cot A = tan(90° − A)
sec A = csc(90° − A)
csc A = sec(90° − A)
For example,
sin 60 0 = cos 30 0 ;
tan 10 0 = cot 80 0 ;
csc 70 0 = sec 20 0 .
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Math 1330 - Section 4.2
Radians, Arc Length, and Area of a Sector
The word trigonometry comes from two Greek roots, trigonon, meaning “having three
sides,” and meter, meaning “measure.” We have already defined the six basic
trigonometric functions in terms of a right triangle and the measures of its three sides.
Before beginning our study of trigonometry, we need to take a look at some basic
concepts having to do with angles.
An angle is formed by two rays that share a common endpoint, called the vertex of the
angle. One ray is called initial side of the angle, and the other side is called the terminal
side. For ease, we typically will draw angles in the coordinate plane with the initial side
along the positive x axis.
We measure angles in two different ways, both of which rely on the idea of a complete
revolution in a circle.
You are probably familiar with degree measure. In this system of angle measure, an
angle which is one complete revolution is 360°. So one degree is
1 th
of a circle.
360
1
The second method is called radian measure. One complete revolution is 2π .
Suppose I draw a circle and construct an angle by drawing rays from the center of the
circle to two different points on the circle in such a way that the length of the arc
intercepted by the two rays is the same as the radius of the circle. The measure of the
central angle thus formed is one radian.
arc length = r
r
θ =1
r
Radian measure of an angle:
Place the vertex of the angle at the center of a circle of radius r. Let s denote the length of
the arc intercepted by the angle. The radian measure θ of the angle is the ratio of the arc
length s to the radius r. That is,
s
r
θ= .
In general, the radian measure of a central angle θ can be determined by the formula
s
r
θ = , where s is the length of the intercepted arc and r is the radius of the circle and r
and s are measured in the same units.
2
Example 1: A circle has radius 12 inches. A central angle θ intercepts an arc of length
36 inches. What is the radian measure of θ?
2π r
= 2π . So the
r
radian measure of the central angle in the case of a complete revolution is 2π .
We know that the circumference of a circle is 2π r . In this case, θ =
Comparing the two systems, then, we have that
2π radians = 360
π radians = 180 °
π
radians = 90 °
2
etc.
As you are becoming more familiar with radian measure, you may find it helpful to be
able to convert between the two systems. We can use the statement π radians = 180 °
to help do this. Dividing both sides of that equation by π , we have that
1 radian =
Similarly,
1 degree =
180°
π
π
180°
so to convert to degrees, multiply by
so to convert to radians, multiply by
180°
π
π
180°
.
.
These are the conversion formulas for radians to degrees and for degrees to radians,
respectively.
3
1 radian =
1 degree =
180°
π
π
180°
so to convert to degrees, multiply by
so to convert to radians, multiply by
180°
π
π
180°
.
.
Example 2: Convert 135° to radian measure.
Example 3: Convert
4π
to degrees.
3
Example 4: Convert
2π
to degrees.
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Example 5: Convert 18° to radian measure.
You will use some angles so often that you should know both their degree and radian
measures. These are:
30° =
45° =
60° =
90° =
π
6
π
4
π
3
π
2
180° = π
360° = 2π
Memorize these!
4
s
r
If θ = , then we can multiply both sides of this equation by r, which gives us
s = rθ .
This is called the arclength formula and it gives the length of the arc intercepted by the
central angle. Note, to use this formula the angle measure MUST be given in radians.
Example 6: If the radius of a circle is 16 inches and the measure of its central angle is
3π
, find the arclength of the sector intercepted by the angle.
4
Example 7: If the arclength of a sector is 8π cm. and the radius is 12 cm., find the
measure of the central angle.
5
A sector of a circle is the region bounded by a central angle and the intercepted arc.
Sometimes, you’ll need to find the area of a sector. The formula for the area of a circle
is A = π r 2 . A sector is a fraction of a circle, determined by the measure of its central
θ
angle over the complete revolution that is a circle, that is
. So the area of a section is
2π
this fraction of the area of the circle, that is:
θ
r 2θ 1 2
A = πr ⋅
=
= r θ.
2π
2
2
2
Note, to use this formula, the measure of the central angle must be given in radians.
Example 8: A sector has radius 10 and central angle measuring 2.5 radians. Find the
area of the sector.
6
Example 9: A sector has central angle measuring 5 radians. The area of the sector is
2500 square units. Find the radius.
Example 10: Find the perimeter of a sector with central angle 60° and radius 3 m.
Example 11: If the area of a sector is 2π m and the measure of the central angle is
π
4
,
find the radius.
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Angular and Linear Velocity
Suppose you are riding on a merry-go-round.
The ride travels in a circular motion, and the horses usually move up and down.
Some of the horses are right along the edge of the merry-go-round, and some are closer to
the center. If you are on one of the horses at the edge, you will travel farther than
someone who is on a horse near the center. But the length of time that both people will
be on the ride is the same. If you were on the edge, not only did you travel farther, you
also traveled faster.
However, everyone on the merry-go-round travels through the same number of degrees
(or radians).
There are two quantities we can measure from this, angular velocity and linear velocity.
The angular velocity of a point on a rotating object is the number of degrees (or radians
or revolutions) per unit of time through with the point turns. This will be the same for all
points on the rotating object.
The linear velocity of a point on the rotating object is the distance per unit of time that
the point travels along its circular path. This distance will depend on how far the point is
from the axis of rotation (the center of the merry-go-round).
We let the Greek letter ω represent angular velocity. Using the definition above,
ω=
θ
t
We denote linear velocity by v. Using the definition above,
v=
a
, where a is the
t
arclength.
The relationship between these two quantities is given by
radius.
v = rω , where r is the
8
ω=
θ
t
v=
a
t
v = rω
Example 12: If the speed of a revolving gear is 25 rpm,
a. Find the number of degrees per minute through which the gear turns.
b. Find the number of radians per minute through which the gear turns.
Example 13: A car has wheels with a 10 inch radius. If each wheel’s rate of turn is 4
revolutions per second,
a. Find the angular speed in units of radians/second.
b. How fast (linear speed) is the car moving in units of inches/second?
9
Example 14: A CD spins at the rate of 500 revolutions per minute. How many degrees
per minute is this?
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Math 1330 - Section 4.3
Unit Circle Trigonometry
An angle is in standard position if its vertex is at the origin and its initial side is along the
positive x axis. Positive angles are measured counterclockwise from the initial side. Negative
angles are measured clockwise. We will typically use the Greek letter θ to denote an angle.
Example 1: Sketch each angle in standard position.
a. 240°
b. -150°
c.
4π
3
d.
−5π
4
1
Angles that have the same terminal side are called coterminal angles. Measures of coterminal
angles differ by a multiple of 360° if measured in degrees or by a multiple of 2π if measured in
radians.
Example 2: Find three angles, two positive and one negative that are coterminal with each
angle.
a. 512°
b.
− 15π
8
2
If an angle is in standard position and its terminal side lies along the x or y axis, then we call the
angle a quadrantal angle.
3
You will need to be able to work with reference angles. Suppose θ is an angle in standard
position and θ is not a quadrantal angle. The reference angle for θ is the acute angle of
positive measure that is formed by the terminal side of the angle and the x axis.
Example 3: Find the reference angle for each of these angles:
a. 123°
b. -65°
c.
7π
9
d.
−2π
3
4
Reference angles:
5
We previously defined the six trigonometric functions of an angle as ratios of the lengths of the
sides of a right triangle. Now we will look at them using a circle centered at the origin in the
coordinate plane. This circle will have the equation x 2 + y 2 = r 2 . If we select a point P(x, y) on
the circle and draw a ray from the origin through the point, we have created an angle in standard
position. The length of the radius will be r.
The six trig functions of θ are defined as follows, using the circle above:
y
r
x
cos θ =
r
y
tan θ = , x ≠ 0
x
sin θ =
r
,y≠0
y
r
sec θ = , x ≠ 0
x
x
cot θ = , y ≠ 0
y
csc θ =
If θ is a first quadrant angle, these definitions are consistent with the definitions given in Section
4.1.
6
An identity is a statement that is true for all values of the variable. Here are some identities that
follow from the definitions above.
sin θ
cos θ
cos θ
cot θ =
sin θ
1
csc θ =
sin θ
1
sec θ =
cos θ
tan θ =
We will work most often with a unit circle, that is, a circle with radius 1. In this case, each
value of r is 1. This adjusts the definitions of the trig functions as follows:
sin θ = y
cos θ = x
tan θ =
y
,x≠0
x
1
, y≠0
y
1
sec θ = , x ≠ 0
x
x
cot θ = , y ≠ 0
y
csc θ =
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Trigonometric Functions of Quadrantal Angles and Special Angles
You will need to be able to find the trig functions of quadrantal angles and of angles measuring
30°, 45° or 60° without using a calculator.
Since sin θ = y and cos θ = x , each ordered pair on the unit circle corresponds to (cos θ , sin θ ) of
some angle θ.
We’ll show the values for sine and cosine of the quadrantal angles on this graph. We’ll also
indicate where the trig functions are positive and where they are negative.
8
Using the identities given above, you can find the other four trig functions of an angle, given just
sine and cosine. Note that some values are not defined for quadrantal angles.
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Values of Trigonometric Functions for Quadrantal Angles
0°
90°
180°
270°
360°
0
π
π
3π
2
2π
2
Sine
Cosine
Tangent
Cotangent
Secant
Cosecant
Example 4: Sketch an angle measuring -270° in the coordinate plane. Then give the six
trigonometric functions of the angle. Note that some of the functions may be undefined.
10
Recall the signs of the points in each quadrant. Remember, that each point on the unit circle
corresponds to an ordered pair, (cosine, sine).
Example 5: Name the quadrant in which both conditions are true:
a. cos θ < 0 and csc θ > 0 .
b. sin θ < 0 and tan θ < 0
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This is a very typical type of problem you’ll need to be able to work.
Example 6: Let P ( x, y ) denote the point where the terminal side of an angle θ intersects the unit
5
circle. If P is in quadrant II and y = , find the six trig functions of angle θ.
13
12
You’ll also need to be able to find the six trig functions of 30°, 60° and 45° angles. YOU MUST
KNOW THESE!!!!!
For a 30° angle:
sin (30°) =
csc(30°) = 2
1
2
cos(30°) =
3
2
tan (30°) =
1
3
sec(30°) =
=
3
3
2
3
=
2 3
3
cot (30°) = 3
For a 60° angle:
sin (60° ) =
3
2
1
cos(60° ) =
2
csc(60°) =
tan (60°) = 3
cot (60°) =
2
3
=
2 3
3
=
3
3
sec(60°) = 2
1
3
13
For a 45° angle:
2
2
2
cos ( 45° ) =
2
tan ( 45° ) = 1
sin ( 45° ) =
csc ( 45° ) = 2
sec ( 45° ) = 2
cot ( 45° ) = 1
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How do we find the trigonometric functions of other special angles?
Method 1: Fill them in. Learn the patterns.
15
Method 2:
The Chart
Write down the angle measures, starting with 0° and continue until you reach 90°. Under these,
write down the equivalent radian measures. Under these, write down the numbers from 0 to 4.
Next, take the square root of the values and simplify if possible. Divide each value by 2. This
gives you the sine value of each of the angles you need. To find the cosine values, write the
previous line in the reverse order.
Now you have the sine and cosine values for the quadrantal angles and the special angles. From
these, you can find the rest of the trig values for these angles. Write the problem in terms of the
reference angle. Then use the chart you created to find the appropriate value.
00
0
3 0°
45°
60°
900
4
2
2
3
3
2
2
0
6
1
2
1
3
2
2
2
1
2
1
3
undefined
0
3
3
3
1
3
3
0
2 3
3
2
2
undefined
2
2
2 3
3
1
Sine
Cosine
Tangent
Cotangent
π
π
undefined
Secant
1
Cosecant
undefined
π
π
1
0
16
Example 7: Sketch an angle measuring 210° in the coordinate plane. Give the coordinates of
the point where the terminal side of the angle intersects the unit circle. Then state the six
trigonometric functions of the angle.
Evaluating Trigonometric Functions Using Reference Angles
1. Determine the reference angle associated with the given angle.
2. Evaluate the given trigonometric function of the reference angle.
3. Affix the appropriate sign determined by the quadrant of the terminal side of the angle in
standard position.
Example 8: Evaluate each:
a. sin(300°)
 3π 
b. tan  
 4 
c. sec (150° )
 −2π 
d. csc 

 3 
 11π 
e. sec

 6 
17
 7π 
f. tan 

 6 
 5π 
g. tan  −

 6 
(
h. tan 240 0
(
)
i. cos − 150 0
)
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UNIT CIRCLE:
19