Download 3.1 Reference Angles Definition of Reference Angle

3.1 Reference Angles (1 of 24)
3.1 Reference Angles
Definition of Reference Angle
The reference angle for any angle θ in standard position is the
positive acute angle formed by the terminal side of θ and the xaxis.
Example 1
Name the reference angle for each.
a. 30
b. 135
c. 240
d. 330
e. −210
f. −140
Reference Angle Theorem
A trigonometric function of an angle and its reference angle are the
same, except, perhaps, for a difference in sign.
3.1 Reference Angles (2 of 24)
Example 1A
Find the sign of all angles that have a reference angle of 30 .
Steps to Find the Trigonometric Functions of Any Angle
1. Find the reference angle θ̂ (read “theta hat”).
2. Determine the sign of the trigonometric function based on the
quadrant that θ terminates in.
3. Write the orginal funciton θ in terms of the same function of
θ̂ .
4.
Find the trigonometric function of θ̂ .
3.1 Reference Angles (3 of 24)
Example 2
Find the exact value of sin 240 . Then
approximate its value to 4 decimal places.
Example 3
Find the exact value of tan 315 . Then
approximate its value to 4 decimal places.
Example 4
Find the exact value of csc 300 . Then
approximate its value to 4 decimal places.
Example 5
Find the exact value of cos 495 . Then
approximate its value to 4 decimal places.
3.1 Reference Angles (4 of 24)
Example 6
Find θ if sin θ = − 0.5592 and θ terminates in
QIII with 0 < θ < 360 .
Example 7
Find θ to the nearest tenth of a degree if
tan θ = − 0.8541 and θ terminates in QIV with
0 < θ < 360 .
Example 8
Find θ if sin θ = −1 / 2 and θ terminates in QIII
with 0 < θ < 360 .
Example 9
Find θ to the nearest degree if sec θ = 3.8637 and
θ terminates in QIV with 0 < θ < 360 .
Example 10
Find θ to the nearest degree if cot θ = −1.6003
and θ terminates in QII with 0 < θ < 360 .
3.2 Radians and Degrees (5 of 24)
3.2 Radians and Degrees
Definition of One Radian
In a circle, a central angle that cuts off an arc equal in length to the
radius of the circle has a measure of 1 radian (rad), as illustrated.
Definition of Radian Measure
If a central angle θ , in a circle of radius r, cuts off an arc length s,
then the measure of θ , in radians is given by s/r. See figure.
3.2 Radians and Degrees (6 of 24)
Example 1
A central angle θ , in a circle of radius 3 cm, cuts
off an arc length of 6 cm. What is the radian measure of θ ?
Notation:
Since θ = s / r , it is a unitless measure. That is,
radians are just real numbers. If no units are shown on an angle,
the units are (by default) radians. Units in degrees must have the
degree symbol. For Example:
θ=2
θ = 2
means the measure of θ is 2 radians
means the measure of θ is 2 degrees
sin 2 = 0.9093 the sine of 2 radians
sin 2 = 0.0349 the sine of 2 degrees
Example 1A Find the angle formed by one full rotation about
the center of a circle of radius r.
3.2 Radians and Degrees (7 of 24)
Radian to Degree Conversion:
2π rad = 360 ,

π rad = 180 ,

π
1 =
rad ,
180

⎛ 180 ⎞
1 rad = ⎜
⎝ π ⎟⎠

One Radian is Much Larger Than One Degree
Examples 2-4
a. 45
Convert each angle to radians. Round to 2
decimal places.
b. 450
c. −78.4 
3.2 Radians and Degrees (8 of 24)
Example 5-7 Convert each angle to degrees.
a. π / 6
b. 4π / 3
c. - 4.5
(round to the nearest tenth of a degree)
Calculator Note
If your calculator is in degree mode you
can convert a radian measure to degree
measure by letting your caclculator
know you entered a radian quantity. For
example, to convert 4π / 3 and -4.5 to
degrees do as shown. The radian
indicator is in the ANGLE menu.
Similary, if your calculator is in radian
mode, then entering 90 will display
1.5708 ≈ π / 2 as the result.
3.2 Radians and Degrees (9 of 24)
Special / Common Angles
Table 1 summarizes displays
the the equivalent radian and
degree measure of the special
angles and the sine, cosine and
tangent of those angles. Figure
12 shows the special angles in
both degrees and radians.
3.2 Radians and Degrees (10 of 24)
π
.
6
Example 8
Find the exact value of sin
Example 9
Find the exact value of 4 sin
Example 10
x = π / 6.
7π
.
6
Find the exact value of 4 sin ( 2x + π ) when
3.2 Radians and Degrees (11 of 24)
Example 11 In navigation, distance is not
measured along a straight line, but along a great
circle due to the Earth roundness. To determine
the great circle distance d (i.e. the arc distance)
between the two points ( LT1 , LN1 ) and
( LT2 , LN 2 ) whose coordinates are given as
latitudes and longitudes is given by
d = sin(LT1 )sin(LT2 ) + cos(LT1 ) cos(LT2 ) cos(LN1 − LN 2 )
The latitudes and longitudes must be in radian measure. Since
most GPS systems give coordinates in degrees and minutes, you
must first convert angles to radians. Find the distance between the
two points with coordinates P1 (N 32 22.108 ′, W 64  41.178 ′ ) and
P2 (N 13 04.809 ′, W 59 29.263′ ) .
3.3 Circular Functions (12 of 24)
3.3 Definition III: Circular Functions
The Unit Circle and Circular Functions
The unit circle is the circle of radius 1
centered at the origin given by
x 2 + y 2 = 1 . The circular functions are
the six trigonometric functions
defined on the unit circle.
Example A
Let (x, y) be any point on the unit
circle and θ be the angle in standard
position whose terminal side
intersects (x, y).
a. Find the six trigonometric
funcitons of θ .
b. Let t be the length of the arc from (1, 0) to (x, y) on the unit circle.
Show that angle θ equals the real number t.
c.
Show that for any point on the unit circle (x, y) = (sin t, cos t).
3.3 Circular Functions (13 of 24)
Definition III: Circular Functions
Let (x, y) be any point on the unit circle and let t be the distance from (1,
0) to (x, y) along the circumference of the unit circle, then
sin t = y
cos t = x
y
tan t = , x ≠ 0
x
x
cot t = , y ≠ 0
y
1
csc t = , y ≠ 0
y
sec t =
1
,x≠0
x
Figure 5 shows the the unit
circle with multiples of π / 6
and π / 4 marked off. The
radian measure of each angle
is the distance from (1, 0) to
the point on the terminal side
of the angle measured
counterclockwise along the
unit circle. The x- and ycoordinates shown are the
cosine and sine, respectively,
of the angle or distance.
Notice that by using reference
angles you only need to
memorize the values of the
circular functions on the
quadrantal angles and the angles that terminate in quadrant QI.
3.3 Circular Functions (14 of 24)
Example 1
Use the unit circle to find the exact values of the six trigonometric
funtions of 5π / 6 .
Example 2
Use the unit circle to find all values of t between 0 and 2π for which
cos t = 1 / 2 .
Example 3
Find the six trigonmetric functions of t if t
correponds to the point (-0.737, 0.675) on
the unit circle.
3.3 Circular Functions (15 of 24)
Example 4
9π
. Identify the function, the argument
4
of the funciton (i.e. the input), and the value of the
function (i.e. the output).
Evaluate sin
Example 5
Evaluate cot 2.37 .
3.3 Circular Functions (16 of 24)
The Circular Functions
Let (x, y) be any point on the unit circle and let t
be the distance from (1, 0) to (x, y) along the
circumference of the unit circle, then
sin t = y
1
csc t = , y ≠ 0
y
cos t = x
sec t =
tan t =
y
,x≠0
x
1
,x≠0
x
x
cot t = , y ≠ 0
y
Example B
Find the domain and range of sin t .
Example C
Find the domain and range of cos t .
3.3 Circular Functions (17 of 24)
Example D
range of tan t .
Find the domain and
Example E
range of cot t .
Find the domain and
Example F
range of csc t .
Find the domain and
Example G
range of sec t .
Find the domain and
3.3 Circular Functions (18 of 24)
The set of all real numbers is denoted (− ∞, ∞) or  . The notation
t ∈ = (− ∞, ∞) is read “t is an element of the real numbers.”
Domain (Input) and Range (Output) of Circular Functions
Let k be any integer.
Function
Domain
Range
t ∈ = (− ∞, ∞)
sin t , cos t
[-1, 1]
(− ∞, ∞)
t ≠ π / 2 + kπ
tan t
(− ∞, 1]∪[1, ∞)
t ≠ π / 2 + kπ
sec t
(− ∞, ∞)
t ≠ kπ
cot t
(− ∞, 1]∪[1, ∞)
t ≠ kπ
csc t
Example 6
number t.
a. cos t = 2
Determine which statements are possible for some real
Example 7
Describe how sec t varies as t increases from 0 to π / 2 .
b. csc π = z
c. tan z = 1000
3.4 Arc Length and area of a Sector (19 of 24)
3.4 Arc Length and Area of a Sector
Arc Length
The definition of the radian measure of θ is
θ = s /r.
So, if θ (in radians) is a central angle in a
circle with radius r, then the length of the arc s
cut off by θ is given by
s = rθ
Example 1
What is the length of the arc
cut off by a central angle of 2 radians in a
circle of radius 4.3 inches.
Example 2
Let the diameter of a Ferris wheel be 250 feet and let
the central angle formed as a rider moves from P0 to P1 be θ . Find the
distance traveled by the rider if θ = 45 , and if θ = 105 .
Example 3
The minute hand of a watch is 1.3 cm long. How far
does the tip of the hand move in 10 minutes?
3.4 Arc Length and area of a Sector (20 of 24)
Arc Length Can Approximation to Chord Length
If the central angle of a circle is realtively small and the radius is relative
large, then the chord length can be approximated by the arc length.
Example A
Suppose the central angle of a circle is θ = 1 and its radius r = 1800 feet
(Figure 5). We will see later on in the course that the chord length is
31.4155 feet (by the Law of Sines). Approximate the chord length with
the arc length.
Example 4
A person standing on the earth observes the length of a
jet flying overhead subtends an angle of 0.45 . If the
length of the jet is 230 feet, approximate the altitude of
the jet.
3.4 Arc Length and area of a Sector (21 of 24)
Area of a Sector
If θ (in radians) is a central angle
in a circle with radius r, then the
area of the sector formed by
angle θ is given by
A=
1 2
rθ
2
Example 5
Find the area of a sector formed
by a central angle of 1.6 radians in a circle of radius 2.1 meters.
Example 6
If the sector formed by a central angle of 15 has an area of π / 3 square
meters, find the exact value of radius of the circle.
Example 7
A lawn sprinkler is located the corner of a yard and is set to rotate 90
degrees and proect water 40.0 feet. How much area is watered by the
spinkler?
3.4 Linear and Angular Velocities (22 of 24)
3.5 Velocities
Definition of Linear Velocity
If P is a point that moves a distance s in time t along the circumference
of a circle, then the linear velocity v of P is given by
s
v=
t
Example 1
A point on a circle travels 5 cm in 2 second. Find the linear velocity of
the point.
Definition of Angular Velocity
If P is a point that moves along a circle to sweep out central angle θ (in
radians) in time t, then the angular velocity ω (omega) of P is given by
θ
ω=
t
Example 2
A point P on a circle rotates through 7π / 6 radians in 5 seconds. Find
the angular velocity of P.
3.4 Linear and Angular Velocities (23 of 24)
Example 3
A wheel with a radius of 17 inches rotates with a velocity of 14 radians
per second. Find the distance traveled by a point on the outer edge of the
wheel.
Example 4
The figure shows a fire truck parked
on the shoulder of a road next to a
long wall. The red light on tip of the
truck is 10 feet from the wall and
rotates through one revolution every
½ second. Find the equation for d
and l in terms of t.
Linear and Angular Velocities
If a point is moving with uniform circular motion on a circle of radius r,
then the linear velocity v and angular velocity ω are relate by: v = ω r
3.4 Linear and Angular Velocities (24 of 24)
Example 5
A phonograph record is turning at 45 rpm. If the distance from the center
of the record to a point on the edge of the record is 4 inches, find the
angular volocity and linear velocity of the point in feet per minute.
Example 6
The Ferris wheel shown has a radius of 250
feet, the bottom of the Ferris wheel is 14 feet
from the ground, and one revolution takes 20
minutes.
a. Find the linear velocity, in miles per hour,
of a person riding on the wheel.
b. The height of the rider in terms of the time
t, where t is measured in minutes.