Unit 5
TRIGONOMETRIC FUNCTIONS OF ANGLES
7.1
ANGLE MEASURE
What is an angle?
 Angle: 2 rays with a common vertex
Positive Angle 
Negative Angle 
 How many angles do I have here?
More about angles:
 An angle is in “standard” position when its initial
side is the positive x-axis and its vertex is at the
origin. Positive angles are formed in a counterclockwise direction and negative angles are formed
in clockwise direction.
What is an angle measured in?
 Radians
 One radian is the measure of the central angle
(Theta-Θ) whose arc (s) is equal to the length of
the radius (r)
What is the circumference of a circle
with a radius of 1 unit?
 2
 So 1 revolution around the coordinate plane = ?
 2
 Therefore 1 revolution around the coordinate plane
= 2 radians
Radians
-3/2
/2

0
-
2
3/2
-/2
Sketch the angle (find coterminal
angles)
 Coterminal angles are angles that have the same




terminal side:
2/3
-/4
7/5
7/3
What is an angle measured in – Part II
 Degrees
 What is the difference between 3 and 3?
 1 = 1/360 of a circle
 Why?
Sketch the angle and find coterminal
angles: Degrees
 150
 282
 -60
 -150
 450
Conversion
 Degrees to Radians
 Multiply by /180
 Radians to Degrees
 Multiply by 180/
Convert the following
 150
 2/3
 282
 -/4
 -60
 7/5
 -150
 7/3
 450
Arc Length –
 s = r  where s is the arc length, r is the radius
and  is the angle in Radians
 If r = 4 in find arc length if  = 240
 If r = 8 in and s = 15 in, find the angle
1 2
Area of a Sector A  r 
2
•What do you think is true about Θ here?
•Find the area of a sector with central
angle 60∘ if the radius of the circle is 3m.
•The area of a sector of a circle with a
central angle of 2 rad is 16 m2. What is the
radius of the circle?
Homework 7.1
 Pg 453-454: 2 – 50 even, 50 – 62 even, 45
7.2a
R I G H T T R I A N G L E T R I G O N O M E T RY
Right Triangle Trigonometry
opp
hyp

adj
Right Triangle Trig:
opp
sin  
hyp
hyp
csc  
opp
adj
cos  
hyp
hyp
sec  
adj
opp
tan  
adj
adj
cot  
opp
SOH CAH TOA
Right Triangle Trigonometry
 Evaluate all 6 trig functions for .
3

4
Sketch the triangle:
 cot  = 5
 cos = 3/7
Special Right Triangle:
/3
/4
/6
/4

Evaluate : sin 30 , cos

6
, tan

3
Solving Triangles – solve for unknowns
15
50
Solving Triangles – solve for unknowns
32
10
Solving Triangles – solve for unknowns
25
12
Homework 7.2a
 Pg 462 – 463: 1 – 4, 11 – 16, 22 – 25, 27, 28
7.2b
R I G H T T R I A N G L E T R I G O N O M E T RY – S TO RY
PROBLEMS
 A kite is held at a 75 angle using 300 ft of string,
how high off the ground is the kite if the person
holds the kite 5 ft off the ground?
x
sin 75 
300

300
x
75
5 ft
 A ladder is placed so that it reaches a point 8
feet from the ground on a wall, if the ladder
makes a 15 with the wall, how long is the ladder?
x
cos 15 
15
8
8
x
 A 40 foot ladder leans against a building. If the
base of the ladder is 6 ft from the base of the
building, what is the angle formed by the ladder
and the building?
 Farmer John looks down into a valley
and sees his favorite cow Eugene
grazing on grass. He notices that a
balloon is floating directly above
Eugene (his favorite cow). Farmer
John determines that the angle of
elevation to the balloon is 72 and
the angle of depression to Eugene is
30, find how far above Eugene the
balloon is if the hill is 40 ft from the
valley below (and Farmer John’s eyes
are 5 feet off the ground)
45
tan30 
x
0
y
tan72 
x
0
72
y
Horizontal line
x
30
45
?
 From a point on the ground 500 ft from the base of
a building, it is observed that the angle of elevation
to the top of the building is 24 degrees, and the
angle of elevation to the top of the flagpole atop
the building is 27 degrees. Find the height of the
building and the length of the flagpole.
Homework 7.2b
 Pg 463 464: 34 – 46 even
7.4
L AW O F S I N E S
H O W D O I S O LV E N O N - R I G H T T R I A N G L E S ?
Laws of Sines – angle and an opposite side
 Given
B
a
C
c
b
sin A sin B sin C


a
b
c
A
a
b
c


sin A sin B sin C
Solve:
B
a
25 
c
20 
C
80.4
A
Solve:
B
a
32
100 
50 
C
b
A
But wait – there is a problem when you know 2
sides and an opposite angle
 What is the sin30?
½
 What is the sin150?
½
 When you do sin-1(1/2) how do you know if its
supposed to be 30 or 150?
Scenario I:
B
7
C
45 
9.9
c
A
Scenario II:
B
248
186
43 
C
b
A
Scenario III:
B
122
C
42 
70
32
A
7.4a Homework
 Pg 483-484: 1, 4, 6, 10, 16, 21, 22
One, Two or Zero?:
A  63, a  17, b  18
A  112, a  32, b  20
A  58, a  10, b  14
7.4b
L A W O F S I N E S – S TO RY P R O B L E M S
Story Problem #1
 The course for a boat race starts at a point A and
proceeds in the direction of S52W to point B, then
in a direction of S40E to point C, and finally back
to A (due North). If A and C are 8 km apart, what is
the total distance for the race?
Story Problem #2
 The pitchers mound on a softball field is 46 feet
from home plate and the distance from the pitchers
mound to first base is 42.6 feet. How far is first
base from home plate?
Story Problem #3
 A hot air balloon is flying above High Point. To the
left side of the balloon, the balloonist measures the
angle of depression to a soccer field to be 20⁰ and
to the right he sees a football field and finds the
angle of depression to be 62.5⁰. If the distance
between the two fields is know to be 4 miles, what
is the direct distance from the balloon to the
soccer field?
 How high is the balloon?
Story Problem #4
 An architect is designing an overhang above a
sliding glass door. During the heat of the summer,
the architect wants the overhang to prevent the
rays of the sun from striking the glass at noon. The
overhang is to have an angle of depression equal to
55⁰ and starts 13 feet above the ground. If the
angle of elevation of the sun during this time is 63⁰,
how long should the architect make the overhang?
Homework 7.4b
 Pg 484 – 485: 24 – 30 even
7.5
L AW O F C O S I N E S
Laws of Cosines – angle and the two included
sides
 Given
a 2  b 2  c 2  2bc  cos A
B
a
C
b  a  c  2ac  cos B
2
c
b
2
2
c  a  b  2ab  cos C
2
A
2
2
Solve:
B
a
212
82 
C
388
A
•Hint always find the smaller angle first
when you use the law of sines after using
the law of cosines – why?
Solve:
B
18
c
47 
C
105
A
Solve:
B
5
C
8
12
A
•Why should we solve for the biggest angle first?
Which is the biggest angle?
Story Problem #1
 A ship travels 60 miles due east, then adjust its
course 15 northward. After traveling 80 miles how
far is the ship from where it departed?
Story Problem #2
 A car travels along a straight road, heading east for
1 hour, then traveling for 30 minutes on another
road that leads northeast. If the car has
maintained a constant speed of 40 mph in what
direction did the car turn?
Homework 7.5
 Pg 491-492: 1-3, 17-21, 24, 25, 27, 32
Navigation Problem/Worksheet
 A boat “A” sights a lighthouse “B” in the direction
N65⁰E and the spire of a church “C” in the direction
S75⁰E. According to a map, the church and
lighthouse are 7 miles apart in a direction of
N30⁰W. Find the bearing the boat should continue
at in order to pass the lighthouse at a safe distance
of 4 miles.
Navigation Worksheet
1
[Type the document title]
Name:
1.
Suppose the lighthouse “B” in the example is sighted at S30⁰W by a ship “P” due north of the
church “C”. Find the bearing “P” should keep to pass “B” at 4 miles distance.
2. In the fog, the lighthouse keeper determines by radar that a boat 18 miles away is heading to
the shore. The direction of the boat from the lighthouse is S80⁰W. What bearing should the
lighthouse keeper radio the boat to take to come ashore 4 miles south of the lighthouse?
3. To avoid a rocky area along a shoreline, a ship “A” travels 7 km to “B”, bearing 22.25⁰, then 8 km
to “C”, bearing 60.5⁰, then 6 km to “D”, bearing 109.25⁰. Find the distance from “A” to “D”.
Unit 5 Test Review