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1
Unit 3 Trigonometry
General Outcome: • Develop trigonometric reasoning.
Specific Outcomes:
3.1 Demonstrate an understanding of angles in standard position, expressed in degrees and
radians.
3.2 Develop and apply the equation of the unit circle.
3.3 Solve problems, using the six trigonometric ratios for angles expressed in radians and
degrees.
3.4 Graph and analyze the trigonometric functions sine, cosine, and tangent to solve
problems. ▪ sin ( )y a b x x d= − +
▪ cos ( )y a b x x d= − +
3.5 Solve, algebraically and graphically, first and second degree trigonometric equations with
the domain expressed in degrees and radians.
3.6 Prove trigonometric identities using:
▪ reciprocal identities
▪ quotient identities
▪ Pythagorean identities
▪ sum or difference identities (restricted to sine, cosine, and tangent)
▪ double-angle identities (restricted to sine, cosine, and tangent)
Topics:
• Trigonometry Fundamentals (Outcomes 3.1 & 3.3) Page 2
• Unit Circle (Outcomes 3.2 & 3.3) Page 24
• Graphing Sine & Cosine Functions (Outcome 3.4) Page 32
• Applications of Sine & Cosine Functions (Outcome 3.4) Page 51
• Graphs of Other Trigonometric Functions (Outcome 3.4) Page 62
• Solving Trigonometric Equations Graphically (Outcome 3.5) Page 66
• Solving Trigonometric Algebraically (Outcome 3.5) Page 70
• Trigonometric Identities (Outcome 3.6) Page 81
• Sum & Difference Identities (Outcome 3.6) Page 91
• Solving Trigonometric Equations Part II (Outcome 3.5) Page 101
2
Unit 3 Trigonometry
Trigonometry Fundamentals:
Standard Position:
An angle is in standard position when its vertex is at the
origin and its initial arm is on the positive x-axis.
Angles in Standard Angles not in Standard
Position: Position:
Angles measured counterclockwise are positive and
angles measured clockwise are negative.
3
Ex) Sketch the following angles.
a) 130 b) 230− c) 490
**Angles that have the same terminal arm are called
Coterminal Angles.
Ex) List 5 angles that are coterminal with 60
**The smallest positive representation of an angle is
called the Principal Angle.
4
Ex) Determine the principal angle for each of the
following.
a) 1040 b) 713− c) 51−
Radian Measure:
Radian measure is a different way in which to measure an
angle. We could measure the distance to Edmonton in km
or miles, meaning we could measure the distance using
two different systems of measurement where 1 km 1
mile. When measuring angles there are 2 different
systems, we can measure angles in degrees or we can
measure them in radians. Note: 1 1 rad .
Radian measure is a ratio of a circles arc length over its
radius
a
r =
5
Ex) What is the measure of an angle that is 360 in
radians?
Ex) Convert the following to radians.
a) 90 b) 45 c) 150
d) 210− e) 212 f) 1080−
rad 180 =
6
Ex) Convert the following to degrees.
a) 3
b)
5
8
c)
22
15
−
d) 6
7
− e) 5.4 f) 11.27−
Ex) Determine the measure of x.
7
Ex) Determine one positive and one negative coterminal
angle for the following.
a) 3
4
b)
7
12
−
Ex) Determine the principal angle in each case.
a) 29
6
b)
11
3
−
8
Special Angles:
Determine the coordinates (height and how far over) of
the sun for each of the following cases.
Thus: sin 45 = & cos45 =
Thus: sin60 = & cos60 =
9
Thus: sin30 = & cos30 =
**Summary**
sin30 = cos30 =
sin 45 = cos45 =
sin60 = cos60 =
10
Quadrants: CAST Rule:
Reciprocal Trigonometric Functions:
opp
sinhyp
= adj
coshyp
= opp
tanadj
=
sin
tancos
=
**New**
csc = sec = cot =
11
Reference Angles:
A reference angle is the angle between the terminal arm
and the horizon or x-axis. Reference angles are always
between 0 and 90 .
Ex) Determine the six exact primary trigonometric ratios
for each of the following.
a) 240
12
b) 4
−
Ex) The point ( )5, 7− lies on the terminal arm of angle
. Determine the six exact primary trigonometric
ratios for .
13
Ex) If 3
sin7
= and cos is negative, determine the other
5 exact primary trigonometric ratios for .
Ex) If 8
sec5
= and tan 0 , determine the exact value of
sin .
14
Trigonometry Fundamentals Assignment:
1) Convert each of the following to radians. Express your answer as an exact
value and as an approximate to the nearest hundredth.
a) 30 b) 300− c) 21
d) 90 e) 750 f) 135−
2) Convert each of the following to degrees. Round your answer to the nearest
hundredth if necessary.
a) 3
b)
5
4
c)
2
3
d) 2.75 e) 21
5
− f) 1
15
3) In which quadrant do each of the following angles terminate?
a) 650 b) 1 c) 192−
d) 11
3
e) 225 f) 8.5
4) Determine one positive and one negative coterminal angle for each of the
following angles.
a) 72 b) 11
7
c) 205−
d) 9.2 e) 520 f) 14
3
16
5) Determine the value of the indicated variable for each of the following cases
below. Round your answers to the nearest hundredth of a unit.
a) b)
c) d)
17
6) A rotating water sprinkler makes one revolution every 15 seconds. The water
reaches a distance of 5 m from the sprinkler.
a) What angle in degrees does the sprinkler rotate through in 9 seconds?
b) What is the area of sector watered in 9 seconds?
7) Complete the table shown below by converting each angle measure to its
equivalent in the other systems. Round your answers to the nearest tenth where
necessary.
Revolutions Degrees Radians
1 rev
270
5
6
1.7−
40−
0.7 rev
3.25− rev
460
3
8
−
18
8) Determine the six primary trigonometric ratios for each of the following. Leave
answers as exact values.
a) 45
b) 240
c) 5
6
d) 3
2
19
9) In with quadrant(s) may terminate under the following conditions?
a) cos 0 b) tan 0 c) sin 0
d) sin 0 e) cos 0 f) sec 0
& cot 0 & csc 0 & tan 0
10) Express the given quantity using the same trigonometric ratio and its reference
angle. For example, cos110 cos70= − . For angle measures in radians, give
exact answers. For example, ( )cos3 cos 3= − − .
a) sin 250 b) tan 290 c) sec135
d) cos4 e) csc3 f) cot 4.95
20
11) Determine the exact value of each expression.
a) cos60 sin30+ b) ( )2
sec45
c) 5 5
cos sec3 3
d) ( ) ( )2 2
tan60 sec60+
e)
2 27 7
cos sin4 4
+
f)
25
cot6
12) Determine the exact measure of all angles that satisfy the following.
a) 1
sin2
−
= , where 0 2
21
b) cot 1 = , where 2 −
c) sec 2 = , where 180 90−
d) ( ) 2cos 1 = , where 360 360−
13) Determine the exact values of the other five trigonometric ratios under the
given conditions.
a) 3
sin5
= , where 2
22
b) 2 2
cos3
−
= , where 3
2
c) 2
tan3
−
= , where 270 360
d) 4 3
sec3
−
= , where 270 180− −
23
14) The point ( )3, 4− lies on the terminal arm of angle . Determine the six exact
primary trigonometric ratios for .
15) The point ( )5, 2− lies on the terminal arm of angle . Determine the six
exact primary trigonometric ratios for .
16) The point ( )5, 12− − lies on the terminal arm of angle . Determine the six
exact primary trigonometric ratios for .
24
Unit Circle:
The unit circle is a circle with a radius of 1 unit whose
center is located at the origin.
Equation of the unit
circle:
2 2 1x y+ =
Consider a point on the unit circle( ), x y to be a point on
the terminal arm of
cos =
sin =
tan =
25
26
Ex) Determine the exact value for the following.
a) sin 240 b) 7
cos4
c) 5
sec6
d) cot540
e) 4
csc3
− f) tan 150−
Ex) Solve the following for x.
a) 2
sin2
x = , where 0 360x
27
b) cos 0.8090x = − , where 0 360x
c) 2
csc3
x−
= , where 0 2x
d) tan x is undefined
e) 24cos 3x = , where 0 2x
28
Unit Circle Assignment:
1) Determine whether or not each of the following points is on the units circle.
a) 3 1
, 4 4
−
b) 5 12
, 13 13
−
c) 5 7
, 8 8
d) 4 3
, 5 5
−
e) 3 1
,2 2
− −
f) 7 3
, 4 4
2) Determine the coordinate for all points on the unit circle that satisfy the
following conditions.
a) 1
, 4
y
in quadrant I b) 2
, 3
x
in quadrant II
29
c) 7
, 8
y−
in quadrant III d) 5
,7
x−
in quadrant IV
e) 1
, 3
x
where 0x f) 12
, 13
y
not in quadrant I
3) If ( )P is the point at the intersection of the terminal arm of angle and the
unit circle, determine the exact coordinates of each of the following.
a) ( )P b) 2
P−
c) 3
P
d) 7
4P
−
e) 5
2P
f) 5
6P
30
4) If is in standard position and 0 2 , determine the measure of if the
terminal arm of goes through the following points.
a) ( )0, 1− b ) 1 1
, 2 2
− −
c) 1 3
, 2 2
d) 3 1
, 2 2
−
e) ( )1, 0 f) 3 1
, 2 2
− −
5) Determine one positive and one negative measure of if 3 1
( ) , 2 2
P −
=
.
31
6) The point 3
( ) , 5
P y
=
lies on the terminal arm of an angle in standard
position and on the unit circle. ( )P is in quadrant IV.
a) Determine the value of y.
b) What is the value of tan ?
c) What is the value of csc ?
32
Graphing Sine and Cosine Functions:
Graph of siny =
• Complete the following table
0 30 60 90 120 150 180 siny x=
210 240 270 300 330 360 siny x=
• Graph siny =
33
Graph of cosy =
• Complete the following table
0
6
3
2
2
3
5
6
cosy x=
7
6
4
3
3
2
5
3
11
6
2
cosy x=
• Graph cosy =
34
Transformations to sin & cos :y x y x= =
sin ( )y a b x c d= − +
Amplitude:
Changing the parameter “a” in sin ( )y a b x c d= − + will
affect the amplitude of the graph.
A = amplitude
*if a is negative, the amplitude is a and the
graph is reflected about the x-axis
35
Period:
Changing the parameter “b” in sin ( )y a b x c d= − + will
affect the period of the graph
360Period
b= or
2Period
b
=
*If b is negative, b is used to find the period of
the graph, the fact that b is negative reflects the
graph about the y-axis.
**Like in Unit 1 the equation must be in the
form sin ( )y a b x c d= − + not sin( )y a bx c d= − +
in order to properly see the period and phase
shift.
36
Phase Shift:
Changing the parameter “c” in sin ( )y b x c d= − + affects
the phase shift of the graph (moves the graph left or
right).
*Think c → determines where we begin
drawing the sine or cosine
pattern
sine begins cosine begins
Vertical Displacement:
Changing the parameter “d” in sin ( )y b x c d= − + affects
the vertical displacement of the graph (moves the graph
up or down).
*Think d → location of the median line
37
Ex) Graph ( )3cos2 120 2y x= − −
Ex) Graph ( )14sin 32 6
y = − +
Ex) Graph ( )4cos3 23
y x = − −
38
Ex) Graph ( )2sin 45 1.5y x= − +
Ex) Graph 4
4cos 23
y x
= −
39
Ex) Determine the equation of a sine function with an
amplitude of 23
and a period of 6
.
Ex) Determine the equation of a cosine function with an
amplitude of 3 and a period of 720 .
Ex) If 3 7
( ) 14sin 504 5
f x
= − +
determine the
following.
a) The maximum of b) The period of
the graph the graph
c) The location of the first minimum found to the
right of the y-axis
40
Ex) Determine both the sine and the cosine equation for
each of the following.
a)
b)
41
c)
d)
42
e)
f)
43
Graphing Sine and Cosine Functions Assignment:
1) Determine the amplitude and the period, in both degrees and radians, for the
graphs of each of the following.
a) ( )2sin 6y = b) 1 1
( ) cos3 3
g x −
=
c) 3
16cos2
y
=
d) 6
( ) 15sin5
f x
= −
2) Determine the range of each function.
a) 3cos 52
y x
= − +
b) ( )2sin 3y x = − + −
d) 1.5sin 4y x= + c) ( )2 3
cos 503 4
y x= + +
44
3) Match each function with its graph.
i) 3cosy = ii) cos3y = iii) siny = − iv) cosy = −
a) b)
c) d)
4) Determine the equation of the cosine curve that has a range given by
6 24,y y y R− and consecutive local maximums at 2
, 249
and
8, 24
9
.
45
5) Match each function with its graph.
i) sin4
y
= −
ii) sin4
y
= +
iii) sin 1y = − iv) sin 1y = +
a) b)
c) d)
6) Determine the equation of the sine curve that has local minimum at ( )15 , 13−
and a local maximum that immediately follows it at ( )35 , 19 . (Note: There
are no other local maximum or minimums between the two given points.)
46
7) Graph each of the following.
a) ( )3
sin 2( 60 ) 24
y = − − .
b) ( )4
4cos 135 23
y
= − +
47
c) 5 2 7
cos 22 3 4
y
= − −
d) 7 1 2 3
sin4 2 3 4
y
= − +
48
8) If ( )y f x= has a period of 6, determine the period of 1
2y f x
=
.
9) If sin 0.3 = , determine the value of ( ) ( )sin sin 2 sin 4 + + + + .
10) Determine both a sine and cosine equation that describes of the graphs given
below.
a)
49
b)
c)
50
d)
11) Determine the value of a to make each statement below true.
a) ( ) ( )4sin 30 4cosx x a− = −
b) ( )2sin 2cos4
x x a
− = −
c) ( )3cos 3sin2
x x a
− − = +
d) ( ) ( )( )cos 2 6 sin 2x x a− + = +
51
Applications of Sine and Cosine Functions:
Ex) By finding the averages of high and low tide, the
depth of water ( )d t in meters, at a sea port can be
approximated using the sine function
( ) 2.5sin 0.164 ( 1.5) 13.4d t t= − +
where t is the time in hours.
a) Sketch the graph of this function.
b) What is the period of the tide (length of time from one
low tide to the next low tide).
c) A cruise ship needs a depth of at least 12 m of water
to dock safely. For how many hours per tide cycle can
the ship safely dock?
52
Ex) The table below shows the average monthly
temperatures for Winnipeg.
Month (m) Jan
1
Feb
2
Mar
3
Apr
4
May
5
June
6
July
7
Aug
8
Sept
9
Oct
10
Nov
11
Dec
12
Temperature (t) -19 -16 -8 3 11 17 20 18 12 6 -5 -14
a) Graph this data.
b) Crate a cosine equation that describes this data.
c) Using the equation created, predict what the
temperature will be on April 15th.
53
Ex) A nail is caught in the tread of a rotating tire at point
N in the following picture.
The tire has a diameter of 50 cm and rotates at 10
revolutions per minute. After 4.5 seconds the nail
touches the ground for the first time.
a) Indicate the proper scale on the horizontal and vertical
axis for the graph above.
b) Determine the equation for the height of the nail as a
function of time.
c) How far is the nail above the ground after 6.5
seconds? (Round your answer to the nearest tenth.)
54
Ex) A ferris wheel has a diameter of 76 m and has a
maximum height of 80 m. If the wheel rotates every
3 minutes, draw a graph that represents the height of
a cart as a function of time. Assume the cart is at its
highest position at 0t = . Show three complete
cycles.
• Determine the cosine equation that describes this
graph.
• How many seconds after the wheel starts rotating
does the cart first reach a height of 10 m? (Round
your answer to the nearest second.)
55
Applications of Sine & Cosine Functions Assignment:
1) The alarm in a noisy factory is a siren whose volume, V decibels fluctuates so
that t seconds after starting, the volume is given by the function
( ) 18sin 6015
V t t
= + .
a) What are the maximum and minimum volumes of the siren?
b) Determine the period of the function.
c) Write a suitable window which can be used to display the graph of the
function.
d) After how many seconds, to the nearest tenth, does the volume first reach
70 decibels?
e) The background noise level in the factory is 45 decibels. Between which
times, to the nearest tenth of a second, in the first cycle is the alarm siren at
a lower level than the background noise?
f) For what percentage, to the nearest per cent, of each cycle is the alarm siren
audible over the background factory noise?
56
2) A top secret satellite is launched into orbit from a remote island not on the
equator. When the satellite reaches orbit, it follows a sinusoidal pattern that
takes is north and south of the equator, (ie. The equator is used as the
horizontal axis or median line). Twelve minutes after it is launched it reaches
the farthest point north of the equator. The distance north or south of the
equator can be represented by the function
( ) 5000cos ( 12)35
d t t
= −
where )(td is the distance of the satellite north or the equator t minutes after
being launched.
a) How far north or south of the equator is the launch site? Answer to the
nearest km.
b) Is the satellite north or south of the equator after 20 minutes? What is the
distance to the nearest km?
c) When, to the nearest tenth of a minute, will the satellite first be 2500 km
south of the equator?
57
3) The height of a tidal wave approaching the face of the cliff on an island is
represented by the equation
2
( ) 7.5cos9.5
h t t
=
where )(th is the height, in meters, of the wave above normal sea level t
minutes after the wave strikes the cliff.
a) What are the maximum and minimum heights of the wave relative to
normal sea level?
b) What is the period of the function?
c) How high, to the nearest tenth of a meter, will the wave be, relative to
normal sea level, one minute after striking the cliff?
d) Normal sea level is 6 meters at the base of the cliff.
i) For what values of h would the sea bed be exposed?
ii) How long, to the nearest tenth of a minute, after the wave strikes the
cliff does it take for the sea bed to be exposed?
iii) For how long, to the nearest tenth minute, is the sea bed exposed?
58
4) The depth of water in a harbour can be represented by the function
( ) 5cos 16.46
d t t
= − +
where )(td is the depth in meters and t is the time in hours after low tide.
a) What is the period of the tide?
b) A large cruise ship needs at least 14 meters of water to dock safely. For
how many hours per cycle, to the nearest tenth of an hour, can a cruise ship
dock safely?
5) A city water authority determined that, under normal conditions, the
approximate amount of water, )(tW , in millions of liters, stored in a reservoir t
months after May 1, 2003, is given by the formula ( ) 1.25 sin6
W t t
= − .
a) Sketch the graph of this function over the next three years.
b) The authority decided to carry out the following simulation to determine if
they had enough water to cope with a serious fire.
“If on November 1, 2004, there is a serious fire which uses 300 000 liters
of water to bring under control, will the reservoir run dry if water rationing
is not imposed? If so, in what month will this occur?
59
6) The graph shows the height, h meters, above the ground over time, t, in
seconds that it takes a person in a chair on a Ferris wheel to complete two
revolutions. The minimum height of the Ferris wheel is 2 meters and the
maximum height is 20 meters.
a) How far above the ground is the person as the wheel starts rotating?
b) If it takes 16 seconds for the person to return to the same height, determine
the equation of the graph in the form ( ) sinh t a bt d= +
c) Find the distance the person is from the ground, to the nearest tenth of a
meter, after 30 seconds.
d) How long from the start of the ride does it take for the person to be at a
height of 5 meters? Answer to the nearest tenth of a second.
60
7) A Ferris wheel ride can be represented by a sinusoidal function. A Ferris wheel
at Westworld Theme Park has a radius of 15 m and travels at a rate of six
revolutions per minute in a clockwise rotation. Ling and Lucy board the ride at
the bottom chair from a platform one meter above the ground.
a) Sketch three cycles of a sinusoidal graph to represent the height of Ling
and Lucy are above the ground, in meters, as a function of time, in
seconds.
b) Determine the equation of the graph in the form ( ) cos ( )h t a b t c d= − + .
c) If the Ferris wheel does not stop, determine the height Ling and Lucy are
above the ground after 28 seconds. Give your answer to the nearest tenth of
meter.
d) How long after the wheel starts rotating do Ling and Lucy first reach 12
meters from the ground? Give your answer to the nearest tenth of a second.
e) How long does it take from the first time Ling and Lucy reach 12 meters
until they next reach 12 meters from the ground? Give your answer to the
nearest second.
61
8) Andrea, a local gymnast, is doing timed bounces on a trampoline. The
trampoline mat is 1 m above ground level. When she bounces up, her feet
reach a height of 3 m above the mat, and when she bounces down her feet
depress the mat by 0.5 m. Once Andrea is in a rhythm, her coach uses a
stopwatch to make the following readings:
• At the highest point the reading is 0.5 seconds.
• At the lowest point the reading is 1.5 seconds.
a) Sketch two periods of the graph of the sinusoidal function which represents
Andrea’s height above the ground, in meters, as a function of time, in
seconds.
b) How high was Andrea above the mat when the coach started timing?
c) Determine the equation of the graph in the form ( ) sinh t a bt d= + .
d) How high, to the nearest tenth of a meter, was Andrea above the ground
after 2.7 seconds?
e) How long after the timing started did Andrea first touch the mat? Answer
to the nearest tenth of a second.
62
Graphs of Other Trigonometric Functions:
Use information about siny = and cosy = to construct
the graph of tany = and coty =
tany = coty =
Period: Period:
Domain: Domain:
Range: Range:
63
Use the knowledge that 1
cscsin
xx
= and 1
seccos
xx
= to
create the graph of cscy x= and secy x= .
Period: Period:
Domain: Domain:
Range: Range:
64
Ex) Determine the period, domain, and range of
3sec 2y x= + .
Ex) Determine the period, domain, and range of
csc3 1y x= − .
Ex) Determine the period, domain, and range of
2tan( 23 )y x= −
65
Graphs of Other Trigonometric Functions Assignment:
1) Determine the Period, Domain, and Range for the graph given by
( )3sec 2 7y = + .
2) Determine the Period, Domain, and Range for the graph given by
1tan 4
3y
= −
.
3) Determine the Period, Domain, and Range for the graph given by
( )2
12csc 40 53
y
= − +
.
4) Identify the restrictions on ( )( )( ) cot 3 60f x = −
66
Solving Trigonometric Equations Graphically:
We can solve equations by graphing each side of the
equation and finding the intersection
Ex) Solve 1
sin2
=
Graph Window
1 sin( )y x= x: [-360, 720, 90]
2 0.5y = y: [-1.5, 1.5, 1]
Interval Notation:
Interval notation gives us another way to represent
solution sets.
Ex)
67
Ex) Solve 24cos 3 = for 0 2 , then state the
general solution.
Ex) Solve 22sin sin 1 0x x+ − = , for 0 360x and state
the general solution.
Ex) Solve 3sec 11 5 + = , for 0 2 , then state the
general solution.
68
Solving Trigonometric Equations Graphically Assignment:
1) Without solving, determine the number of solutions for each trigonometric
equation in the specified domain. Explain your reasoning.
a) 3
sin2
= , 0 2 b) 1
cos2
= , 2 2 −
b) tan 1 = − , 360 180− d) 2 3
sec3
= , 0 2
2) The equation 1
cos2
= , 0 2 , has solutions 3
and
5
3
. Suppose the
domain is not restricted.
a) What is the general solution corresponding to 3
= ?
b) What is the general solution corresponding to 5
3
= ?
69
3) Solve each equation for 0 2 . Give solutions to the nearest hundredth of
a radian.
a) tan 4.36 = b) sin 0.91 =
c) sec 2.77 = d) csc 1.57 = −
4) Solve each equation in the specified domain.
a) 3cos 1 4cos − = , 0 2
b) 2 sin 1 0x − = , 360 360x−
c) 3 tan 1 0 + = , 2 −
d) 3sec 2 0x + = , 3 −
5) Explain why the equation sin 0 = has no solution in the interval ( ), 2 .
70
Solving Trigonometric Equations Algebraically:
When solving trigonometric equations we must remember
that there are usually 2 principal solutions (2 solutions
within 360 or 2 ).
Tools for Solving:
S A 1
sin302
= 3
cos302
=
T C 2
sin 452
= 2
cos452
=
3
sin602
= 1
cos602
=
sin
tancos
=
coscot
sin
=
1csc
sin
=
1sec
cos
=
*understand how reference angles can be used
siny x= cosy x=
71
Level I:
Isolate the trigonometric function, then use your
knowledge of reference angles to find solutions
Ex) Solve each of the following.
a) 3
sin2
= , 0 360 and the general solution
b) 2cos 1 0 − = , 0 2 and the general
solution
c) 2sin 2 0 + = , )0, 2 and the general
solution
72
d) 1 cos 3cos + = , )0 , 360 and the general
solution
e) 24sin 2 5 + = , 0 2 and the general
solution
f) 4csc 3 5 + = − , )0 , 360 and the general
solution
73
Level II:
Solve by factoring
Ex) Solve the following.
a) 2cos cos 0x x− = , 0 360x and the general
solution
b) sin tan sinx x x= , 0 2x and the general
solution
c) cos csc 2cos 0 + = , )0, 2 and the general
solution
74
d) 22sin sin 1 0x x− − = , )0 , 360 and the general
solution
e) 22sin 7sin 4x x+ = , )0, 2 and the general
solution
f) 2csc 3csc 28 0 − − = , 0 360 and the general
solution
75
Level III:
These involve multiple angle equations
Ex) 1
sin 22
x = , 2cos3 2 = , 2 1tan 1 02
x − =
Solve for 2x , or 3 , or 12
x , etc , listing all principle
solutions and general solution then give answers for just x
or .
Ex) Solve the following.
a) cos2 1 0 + = , 0 360 and the general
solution
76
b) 1
sin 22
x−
= , )0, 2 and the general solution
c) 12sin 32
x = , 0 2x and the general
solution
d) sec3 2 = − , )0 ,360 and the general solution
77
Solving Trigonometric Equations Algebraically Assignment:
1) Solve the following Level I equations. In each case, provide answers for the
specified domain and provide the general solution.
a) 3
sin2
x = , 0 360x b) tan 1x = − , 0 2x
c) 2cos 2 0 − = , 2x − d) 3cot 7 6 + = , 0 360x
e) 5sin 3sin 1x x= − , )0 , 360 f) 3sec 1 1x − = , )0, 2
g) 25csc 3 13 + = , )0, 2 h) 7cos 4 3 − = , )0 , 540
78
2) Solve the following Level II equations. In each case, provide answers for the
specified domain and provide the general solution.
a) 2sin sin 0x x− = , )0 , 360
b) 2sin cos cos 0 + = , )0, 2
c) tan sec tan 0x x x− = , 0 2x
d) 22cos 3cos 1 0 − + = , 0 360
79
e) 2tan tan 2 0x x− − = , )0, 2
f) 2sec 2sec 3 0 − − = , )0 , 360
g) 22csc 2 3csc − = , 0 360
h) 26sin 5sin 1x x− = − , 0 2x
80
3) Solve the following Level III equations. In each case, provide answers for the
specified domain and provide the general solution.
a) 3
sin 22
x = , 0 360x
b) 2cos3 2 + , 0 2
c) cot 2 1 0x − = , )0 , 360
d) sin3 cos3 sin3 0 + = , )0, 2
e) 2 1 12sin sin 1 0
2 2x x
− − =
, 0 2x
81
Trigonometric Identities:
We already know some basic trigonometric identities:
New identities:
Proof:
1csc
sin
=
1sec
cos
=
sintan
cos
=
1 coscot
tan sin
= =
2 2sin cos 1 + = 2 2tan 1 sec + =
2 21 cot csc + =
82
Strategies for proving identities:
• Convert everything so it is written in terms of sine or
cosine
• Simplify
• Try to make one side of the identity match the other
** Remember: We must treat each side of the identity as
independent of one another. We can not
treat these like an equation.
Ex) Prove the following identities.
a) sin cot cos cscx x x x+ =
83
b) 2
2tan2sin cos
1 tan
AA A
A=
+
c) 21 12csc
1 cos 1 cosx
x x+ =
+ −
84
d) cos 1 sin
1 sin cos
x x
x x
+=
−
e) Verify the identity sin cot cos cscx x x x+ = for 6
x
=
85
Trigonometric Identities Assignment:
1) Determine the non-permissible values of x, in radians, for each expression.
a) cos
sin
x
x b)
sin
tan
x
x
c) cot
1 sin
x
x− d)
tan
cos 1
x
x +
2) Simplify each expression to a single trigonometric function sin x , cosx ,
tan x , cscx , secx , or cot x .
a) sec sinx x b) 2sec cot sinx x x c)
cos
cot
x
x
d) cos tan
tan sin
x x
x x
e) csc cot sec sinx x x x f) 2
cos
1 sin
x
x−
86
3) Verify that the equation sec
sintan cot
xx
x x=
+ is true for
4x
= .
4) Verify that the equation sin cos 1 cos
1 cos tan
x x x
x x
−=
+ is true for 30x = .
5) Compare siny x= and 21 cosy x= − by completing the following.
a) Verify that 2sin 1 cosx x= − is true for 60x = , 150x = , and 180x = .
b) Graph siny x= and 21 cosy x= − in the same window.
c) Determine whether 2sin 1 cosx x= − is an identity. Explain your answer.
87
6) Simplify ( ) ( )2 2
sin cos sin cosx x x x+ + − .
7) Determine an expression for m that makes 22 cos
sinsin
xm x
x
−= + an identity.
8) Prove the following identities and state any restrictions that may apply.
a) 21 12sec
1 sin 1 sinx
x x+ =
+ −
88
b) 2
2
sin sin cossin
sin
x x xx
x
−=
c) 2
sin cos sincsc cot
cos 1
−= −
−
d) 2cos cos tan sec + =
89
e) 2 2sin cos
sin cossin cos
x xx x
x x
−= −
+
f) 2
2
1 sin 1 sin
1 2sin 3sin 1 3sin
x x
x x x
− +=
+ − +
g) sin cos cot
1 cos sin 1 cos
+ =
+ +
90
h) cos cos
cotsec 1 sec 1
x xx
x x+ =
− +
i) sin cos cot csc + =
j) sin cos 1 cos
1 cos tan
−=
+
91
Sum and Difference Identities:
Ex) Use the above identities to write the following as a
single trigonometric function.
a) sin20 cos32 sin32 cos20−
b) cos15 cos30 sin15 sin30−
( )sin sin cos sin cosA B A B B A+ = +
( )sin sin cos sin cosA B A B B A− = −
( )cos cos cos sin sinA B A B A B+ = −
( )cos cos cos sin sinA B A B A B− = +
tan tantan( )
1 tan tan
A BA B
A B
++ =
−
tan tantan( )
1 tan tan
A BA B
A B
−− =
+
( )sin 2 2sin cosA A A= ( ) 2 2cos 2 cos sinA A A= −
2
2 tantan(2 )
1 tan
AA
A=
−
2cos(2 ) 2cos 1A A= −
2cos (2 ) 1 2sinA A A= −
92
Ex) Determine the exact value of
sin75 cos15 cos75 sin15−
Ex) Use the sum and difference identities to determine
the exact value of the following.
a) cos15 b) 2
sin3
93
c) 2 2cos 15 sin 15− d) 5
tan12
Prove the following trigonometric identities.
a) cos( )cos sin( )sin cosx y y x y y x+ + + =
94
b) sin( )
1 cot tansin cos
x yx y
x y
++ =
c) 2 2sin( )sin( ) cos cosx y x y y x+ − = −
95
Sum and Difference Identities Assignment:
1) Write each expression as a single trigonometric function.
a) cos43 cos27 sin 43 sin 27− b) sin15 cos20 cos15 sin 20+
c) 2 2cos 19 sin 19− d) 3 5 3 5
sin cos cos sin2 4 2 4
−
e) 8sin cos4 4
f)
2
2tan 76
1 tan 76−
g) 21 2cos12
− h) ( )2 26cos 24 6sin 24 tan 48−
2) Simplify and then give the exact value for each expression.
a) cos40 cos20 sin 40 sin 20− b) sin 20 cos25 sin 25 cos20+
c) 2 2cos sin6 6
− d) cos cos sin sin
2 3 2 3
−
96
3) Simplify ( )cos 90 x− using a difference identity.
4) Determine the exact value for each of the following.
a) cos75 b) tan165
c) 7
sin12
d) sec195
97
5) Angle is in quadrant II and 5
sin13
= . Determine an exact value for each of
the following.
a) cos2 b) sin 2
6) If the point ( )2, 5 lies on the terminal arm of angle in standard position,
determine the value ( )cos + .
7) What value of k makes the equation sin5 cos cos5 sin 2sin cosx x x x kx kx+ =
true?
98
8) If A and B are both in quadrant I, and 4
sin5
A = and 12
cos13
B = , evaluate
each of the following.
a) ( )cos A B− b) ( )sin A B+
9) Prove the following identities and state any restrictions that may apply.
a) 4 4cos sin cos2x x x+ =
b) 21 cos2sin
2
xx
−=
99
c) 24 8sin 4
2sin cos tan 2
−=
d) csc
csc22cos
xx
x=
e) 2
sin tan sin2
1 cos 2cos
+=
+
100
f) sin2 cos2
csccos sin
x xx
x x+ =
g) ( ) ( )sin 90 sin 90 + = −
h) sin4 sin2
tancos4 cos2
x xx
x x
−=
+
101
Solving Trigonometric Equations Algebraically: (Part 2):
Type IV:
These equations first require you to make a substitution
using a trigonometric identity so that the equation
becomes a Type I, II, or III question.
Ex) Solve the following.
a) cos2 1 cos 0x x+ − = , )0 ,360 and the general
solution
b) 21 cos 3sin 2x x− = − , 0 2x and the general
solution
102
c) sin2 2 cosx x= , )0, 2 and the general
solution
d) 2sin 7 3cscx x= − , 0 360 and the general
solution
103
Solving Trigonometric Equations Algebraically Part II Assignment:
Solve the following equations. In each case, provide answers for the specified
domain and provide the general solution.
1) sin2 sin 0x x− = , 0 360x
2) cos2 sinx x= , )0, 2
3) cos cos2 0 − = , 0 2
4) tan cos sin 1 0 − = , )0 , 360
104
5) cos2 3sin 2x x− = , )0, 2
6) 3csc sin 2 − = , 0 360x
7) 2 2sin cos 1x x= + , 0 2
8) sin2 2cos cos2x x x= , )0 , 360
105
Answers: Trigonometry Fundamentals Assignment:
1. a) 6
, 0.52 b)
5
3
−, 5.24− c)
7
60
, 0.37 d)
2
, 1.57
e) 25
6
, 13.09 f)
3
4
−, 2.36
2. a) 60 b) 225 c) 120 d) 157.56 e) 756− f) 57.30
3. a) 4 b) 1 c) 2 d) 4 e) 3 f) 2
4. Possible answers could be: a) 288− , 432 b) 3
7
−,
25
7
c) 565− , 155 d) 3.37− , 2.92 e) 200− , 160 f) 4
3
−,
2
3
5. a) 2.25 = or 432 = b) 3.82r = cm c) 17.10a = m
d) 10.98a = ft
6. a) 6
5
or 3.77 or 216 b) 47.12 2m
7.
Revolutions Degrees Radians
1 rev 360 2
3
4 rev
270 3
2
5
12 rev
150 5
6
0.3− rev 97.4− 1.7−
1
9 rev
40− 2
9
−
0.7 rev 252 7
5
3.25− rev 1170− 13
2
−
23
18 rev
460 23
9
3
16 rev
67.5− 3
8
−
106
8. a) 2
sin 452
= , 2
cos452
= , tan 45 1= , csc45 2= , sec45 2= ,
cot 45 1=
b) 3
sin 2402
−= ,
1cos240
2
−= , tan 240 3= ,
2csc240
3
−= ,
sec240 2= − , 1
cot 2403
=
c) 5 1
sin6 2
= ,
5 3cos
6 2
−= ,
5 1tan
6 3
−= ,
5csc 2
6
= ,
5 2sec
6 3
−= ,
5cot 3
6
= −
d) 3
sin 12
= − ,
3cos 0
2
= ,
3tan undefined
2
= ,
3csc 1
2
= − ,
3sec undefined
2
= , cot 45 1=
9. a) 1 & 4 b) 2 & 4 c) 1 & 2 d) 2 e) 2 f) 1
10. a) sin 250 sin70= − b) tan 290 tan70= − c) sec135 sec45= −
d) ( )cos4 cos 4 = − − e) ( )csc3 csc 3= − f) ( )cot 4.95 cot 2 4.95= − −
11. a) 1 b) 2 c) 1 d) 7 e) 1 f) 3
12. a) 7 11
, 6 6
= b)
3 5, ,
4 4 4
−= c) 60 , 60 = −
d) 360 , 180 , 0 , 180 = − −
13. a) 4
cos5
−
= , 3
tan4
−
= , 5
csc3
= , 5
sec4
−
= , 4
cot3
−
=
b) 1
sin3
−
= , 1
tan2 2
= , csc 3 = − , 3
sec2 2
−
= , cot 2 2 =
c) 2
sin13
−
= , 3
cos13
= , 13
csc2
−
= , 13
sec3
= , 3
cot2
−
=
d) 13
sin4
= , 3
cos4 3
−
= , 39
tan3
−
= , 4
csc13
= , 3
cot39
−
=
14. 4
sin5
= , 3
cos5
−
= , 4
tan3
−
= , 5
csc4
= , 5
sec3
−
= , 3
cot4
−
=
107
15. 2
sin29
−
= , 5
cos29
= , 2
tan5
−
= , 29
csc2
−
= ,
29sec
5 = ,
5cot
2
−=
16. 12
sin13
−
= , 5
cos13
= , 12
tan5
−
= , 13
csc12
−
= , 13
sec5
= ,
5cot
12
−=
Unit Circle Assignment:
1. a) No b) Yes c) No d) Yes e) Yes f) Yes
2. a) 1 15
, 4 4
b) 5 2
, 3 3
−
c) 7 15
, 8 8
− −
d) 2 6 5
, 7 7
− −
e) 2 2 1
, 3 3
−
f) 12 5
, 13 13
−
3. a) ( )1, 0− b) ( )0, 1− c) 1 3
, 2 2
d) 2 2
, 2 2
e) ( )0, 1 f) 3 1
, 2 2
−
4. a) 3
2
= b)
5
4
= c)
3
= d)
5
6
= e) 0 =
f) 7
6
=
5. 150 or 5
6
& 210− or
7
6
−
6. a) 4
5y
−= b)
4tan
3
−= c)
5csc
4
−=
Graphing Sine & Cosine Functions Assignment:
1. a) Amplitude 2= , Period 60 or 3
= b)
1Amplitude
3= ,
108
Period 1080 or 6= c) Amplitude 16= , 4
Period 240 or 3
=
d) Amplitude 15= , 300 5
Period or 3
=
2. a) 2 8,y y y R b) 5 1,y y y R− −
c) 2.5 5.5,y y y R d) 1 17
,12 12
y y y R
3. a) i b) iv c) iii d) ii
4. 2
15cos 3 99
y
= − +
5. a) iv b) ii c) iii d) i
6. ( )( )16sin 9 25 3y = − +
7. a)
b)
109
c)
d)
8. 12
9. 0.9
10. a) ( )1
2sin 902
y = − + , ( )1
2cos 1802
y = − +
b) 7 1
sin 14 2 6
y
= + +
, 7 1 5
cos 14 2 6
y
= − +
c) 3 4 3
sin2 3 4 2
y
= − +
, 3 4 5 3
cos2 3 8 2
y
= − +
d) 5 1
sin4 6 4
y
= − −
, 5 2 1
cos4 3 4
y
= − −
11. a) 120 b) 3
4
c) 2 d)
15
4
110
Applications of Sine & Cosine Functions Assignment:
1. a) Maximum 78 decibels= , Minimum 42 decibels= b) 30 sec.
c) : 0, 9, 10x : 0, 80, 10y (shows 3 cycles) d) 2.8 sec.
e) 19.7 sec. & 25.3 sec. f) 81%
2. a) 2369 km North b) 3765 km North c) 35.3 min.
3. a) Maximum 7.5 m= , Minimum 7.5 m= − b) 9.5 min. c) 5.9 m
d) i) 6 mh − ii) 3.8 min. iii) 1.9 min.
4. a) 12 hours b) 7.9 m
5. a)
b) Yes it will run dry in July of 2004.
6. a) 11 m b) 9sin 1116
h t
= +
c) 4.6 m d) 28.3 sec.
7. a)
b) ( )( ) 15cos 5 165
h t t
= − +
c) 11.4 m d) 2.1 sec.
e) 6 sec.
111
8. a)
b) 1.75 m c) ( ) 1.25sin 1.75h t t= + d) 2.8 m e) 1.2 sec.
Graphs of Other Trigonometric Functions Assignment:
1. Period 360= , Domain: 45 90 , ,n n I R + ,
Range: 4 10,y y y R
2. Period 540= , Domain: 270 540 , ,n n I R + , Range: y y R
3. Period 540= , Domain: 40 270 , ,n n I R + ,
Range: 7, 17,y y y y R −
4. 60 , ,n n I R
Solving Trigonometric Equations Graphically Assignment:
1. a) 2 b) 4 c) 3 d) 2
2. a) , 3
n n I
= + b) 5
, 3
n n I
= +
3. a) 1.35, 4.49 = b) 1.14, 2.00 = c) 1.20, 5.08 =
d) 3.83, 5.59 =
4. a) 0 = b) 315 , 225 , 45 , 135x = − − c) 5 11
, , 6 6 6
−=
d) 5 5 7 17
, , , 6 6 6 6
x −
=
5. sin 0 = has solutions of 0, , and 2 , but none of these are included in the
interval ( ), 2 .
112
Solving Trigonometric Equations Algebraically:
1. a) 60 , 120x = ; 60 360
, 120 360
nx n R
n
+=
+
b) 3 7
, 4 4
x
= ; 3
, 4
x n n R
= +
c) 7
, ,4 4 4
−= ;
24
, 7
24
n
n R
n
+
= +
d) 120 , 300 = ; 120 180 , n n R = +
e) 210 , 330x = ; 210 360
, 330 360
nx n R
n
+=
+
f) 11
, 6 6
x
= ;
26
, 11
26
n
x n R
n
+
= +
g) 3 5 7
, , , 4 4 4 4
= ; ,
4 2n n R
= +
h) 0 , 360 = ; 360 , n n R =
2. a) 0 , 90 , 180x = ; 180
, 90 360
nx n R
n
=
+
b) 7 3 11
, , , 2 6 2 6
= ;
2
2 3 ,
32
2
n
n R
n
+
= +
c) 0, x = ; , x n n R=
d) 0 , 60 , 300 = ;
360
60 360 ,
300 360
n
n n R
n
= +
+
113
e) 3 7
1.107, , 4.25, 4 4
x
= ;
1.107
, 3
4
n
x n Rn
+
= +
f) 70.5 , 180 , 289.5 = ;
70.5 360
180 360 ,
289.5 360
n
n n R
n
+
= +
+
g) 30 , 150 = ; 30 360
, 150 360
nn R
n
+=
+
h) 0.34, 0.52, 2.62, 2.80x = ;
0.34 2
0.52 2
2.62 2
2.80 2
n
nx
n
n
+
+=
+ +
3) a) 30 , 60, 210 , 240x = ; 30
, 60
nx n R
n
+=
+
b) 5 11 13 19 7
, , , , ,4 12 12 12 12 4
= ;
2
4 3 ,
5 2
12 3
n
n R
n
+
= +
c) 22.5 , 112.5 , 202.5 , 292.5x = 22.5 90 , x n n R= +
d) 2 4 5
0, , , , ,3 3 3 3
= ; ,
3n n R
=
e) x = ; 4
, 3
x n n R
= +
Trigonometric Identities Assignment:
1. a) x n b) 2
x n
c) x n , 22
x n
+
d) 2
x n
+ , 2n +
2. a) tan x b) sin x c) sin x d) cot x e) cscx f) secx
114
Sum & Difference Identities Assignment:
1. a) cos70 b) sin35 c) cos38 d) sin4
e) 4sin
2
f) tan152 g) cos6
− h) 6sin 48
2. a) 1
2 b)
2
2 c)
1
2 d)
1
2
−
3. sin x
4. a) 6 2
4
− b) 3 2− c)
6 2
4
+ d)
4
6 2+
5. a) 119
169 b)
120
165
6. 2
29
−
7. 3k =
8. a) 56
65 b)
63
65
Solving Trigonometric Equations Algebraically Part II Assignment
1. 0 , 60 , 120 , 180x = ,
180
60 360 ,
120 360
n
x n n I
n
= +
+
2. 5 3
, , 6 6 2
x
= ; 2
, 6 3
x n n I
= +
3. 2 4
0, , 3 3
= ;
2 ,
3n n I
=
4. 90 , 270 = ; 90 180 , n n I = +
115
5. 7 3 11
, , 6 2 6
x
= ;
72
6
32 ,
2
112
6
n
x n n I
n
+
= +
+
6. 90 = ; 90 360n = +
7. 3
, 2 2
= ; ,
2n n I
= +
8. 30 , 90 , 150 , 270x = ; 30 120
, 90 360
nx n I
n
+=
+
