Review
Chapter 6 (Trigonometric
Functions of Angles)
Angle Measure:
Degrees:
The measure of an angle is the amount of rotation about the
vertex required to move R1 onto R2 , this is measured in degrees.
An angle of measure 1 degree is formed by rotating the initial side
1
of a complete rotation. In calculus and other branches of
360
mathematics, angle opens is measured along the arc of a circle of
radius 1 with its center at the vertex of the angle.
Radian Measure: If a circle of radius 1 is drawn with the vertex of an
angle at its center, then the measure of this angle in radians is the
length of the arc that subtends the arc.
Angle in Standard Position: An angle is in standard position if it is drawn in the
xy-plane with its vertex at the origin and its initial side on the positive xaxis.
Relationship between Degrees and Radians:
180   rad
 180 
1 rad  

  

1 
Converting degrees to radians:
multiply by
Converting radians to degrees:
multiply by

180
rad

180
180

Examples of problems with Relationship between Degrees and Radians:
(1)
Convert the following degree measures to radian measures:
(a)
60 degrees (b)
(d)
20 (c)
135
7.5
(2)
Convert the following radian measures to degree measures:

3


(a)
(b)
(c)
3
(d)
6
4
9
Coterminal Angles:
Two angles whose terminal sides coincide.
To find coterminal angles (degrees):
 360 to given angle (can be
done multiple times)
To find coterminal angles (radians):
 2 to given angle (can be done
multiple times)
Examples of problems with Coterminal Angles:
(1)
List one positive and one negative angle that is coterminal with each given
12
3

angle: (a)
(b)
(d)
 320
640 (c)
5
4
(2)
Determine if the following pairs of angles are coterminal:
21
45
(a)
(b)
and
 455 and 990
4
4
Length of a Circular Arc:
In a circle of radius r, the length s of an arc that subtends a central angle
s
of  radians is
s  r or  
r
Examples of problems with Length of a Circular Arc:
(1)
(2)
Find the length of an arc of a circle with radius 10 m that subtends a
central angle of 30 .
A central angle  in a circle of radius 9 m is subtended by an arc of length
12 m. Find the measure of  in radians.
Area of a Circular Sector:
In a circle of radius r, the area A of a sector with a central angle of 
1
A  r 2
radians is
2
Examples of problems with Area of a Circular Sector:
(1)
Find the area of a sector of a circle with central angle 60 if the radius of
the circle is 3 m.
(2)
Find the area of a sector of a circle with central angle of 4  if the radius of
the circle is 45 meters.
Linear and Angular Speeds:
Suppose a point moves along a circle of radius r and the ray from the
center of the circle to the point traverses  radians in time t . Let s  r
be the distance the point travels in time t . Then the speed of the object is
given by

w  (THINK ANGLE PER PERIOD OF TIME)
Angular Speed
t
s
Linear Speed v 
where s  r (THINK DISTANCE PER PERIOD OF TIME)
t
Examples of problems with Linear and Angular Speeds:
(1)
A disk with a 12 inch diameter spins at the rate of 45 revolutions per
minute. Find the angular and linear velocities of a point at the edge of the
disk in radians per second and inches per second, respectively.
(2)
A boy rotates a stone in a 3 foot long sling at the rate of 15 revolutions
every 10 seconds. Find the angular and linear velocities of the stone.
Relationship between Linear and Angular Speed:
If a point moves along a circle of radius r with angular speed w , then its
linear speed v is given by
v  rw
(THINK DISTANCE PER PERIOD OF TIME)
Examples of problems with Relationship between Linear and Angular Speed:
(1)
A woman is riding a bicycle whose wheels are 30 inches in diameter. If
the wheels rotate at 150 revolutions per minute, find the speed at which
she is driving.
(2)
A man is riding a bicycle whose wheels are 26 inches in diameter. If the
wheels rotate at 125 revolutions per minute (rpm), find the speed at which
she is traveling in miles per hour.
Circles Attached by Chain vs Pole
If two different sized circles are attached by chain the rpm’s are different
If two different sized circles are attached by a pole the rpm’s are the same
Examples of problems with Circles Attached by Chain vs Pole
Two wheels, A and B, are connected by a chain. If wheel A has a diameter of 20
inches and rotates and 366 times per minute, find:
a)
the angular speed of wheel A
b)
If the diameter of wheel B is 80 inches, find its rpm.
c)
what is the angular speed of wheel B in this case in radians per
minute?
d)
If the diameter of wheel B is three times as big as wheel A, find its
rpm.
e)
what is the angular speed of wheel B in this case in radians per
minute?
f)
If the diameter of wheel B is 10 inches, find its rpm.
g)
what is the angular speed of wheel B in this case in radians per
minute?
The Trigonometric Ratios
opposite
sin  
hypotenuse
hypotenuse
csc  
opposite
Remember
csc  
opposite
adjacent
adjacent
cot  
opposite
adjacent
hypotenuse
hypotenuse
sec  
adjacent
cos  
1
sin 
sec  
tan  
1
cos 
Pythagorean Identities:
sin 2   cos 2   1
tan 2   1  sec 2 
cot  
1
tan 
1  cot 2   csc 2 
Signs of the Trigonometric Functions:
y
Sine
All
x
Tangent
Cosine
Finding Trigonometric Functions: When given a trig ratio you can use
either geometric method or Pythagorean identities to find all other
trig functions.
Examples of problems with The Trigonometric Ratios:
(1)
Find the six trigonometric ratios of the angle  in the following
diagram.
3
2

5
(2)
Consider a right triangle with  as one of its acute angles. If cos  
find the other five trigonometric ratios of  .
7
,
8
(3)
(4)
(5)
(6)
(7)
(8)
2
and  is in quadrant II all other trig functions.
3
If sec  6 and  is in quadrant IV, find all the other trig functions.
What is the value of:
(a)
(b)
(c)
cos17
sec 30
tan 23
A giant redwood tree casts a shadow 452 ft long. Find the height of the
tree if the angle of elevation of the sun is 12.3 .
From a point on the ground 700 ft from the base of a building, it is
observed that the angle of elevation to the top of the building is 21 and
the angle of elevation to the top of a flagpole atop the building is 23 .
Find the height of the building and the length of the flagpole.
A 50 foot ladder leans against a building. If the base of the ladder is 7 ft
from the base of the building, what is the angle formed by the ladder and
the building?
If tan  
Values of the trigonometric ratios for special triangles:
sin 
cos
tan 
csc 
 in
 in
degrees radians
30

1
3
3
2
6
2
2
3
45

2
2
1
2
4
2
2
60

1
3
2 3
3
2
3
2
3
sec 
cot 
2 3
3
3
2
1
2
3
3
The previous chart can also be used to find the points of intersection of the
terminal side of an angle and the unit circle after a rotation of  degrees/radians
 (radians, degrees)
Terminal point determined by 
0, 0

, 30
6
(1,0)
 3 1


 2 ,2



, 45
4
 2 2


 2 , 2 



, 60
3
1 3
 ,

2 2 



, 90
2
(0,1)
Reference Angle:
Let  be an angle in standard position. The reference angle 
associated with  is the acute angle formed by the terminal side of
 and the x-axis.
Evaluating Trigonometric Functions for Any Angle:
To find the values of the trigonometric functions for any angle  ,
we carry out the following steps.
1)
Find a coterminal angle between 0 and 360 degrees or 0
and 2 radians
2)
Use your chart to find the trig function of the new angle
Examples of problems with Values of the trigonometric ratios for special
triangles:
(1)
Find the reference angle for:
5
17
(a)  
(b)   870
(c)  
(d)   535
3
6
(2)
Find (a) sin 240 (b) cot 495 (c) sec 480 (d) cos 810
16
43
 17 
(e) csc 570 (f) tan  495 (g) sin
(h) sec 
(i) sin

3
6
6 

55
19
22
(j) cos
(k) sin 
(l) sin
6
4
3
Area of a Triangle: The area of a triangle with sides of length a , b and with
1
A  ab sin 
included angle  is:
2
Examples of problems with Area of a Triangle:
(1)
Find the area of triangle ABC shown in the following diagram:
B
3 cm
A
120
10 cm
C