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SESSION 5
Trig. Functions and Graphs
Math 30-1
3
R
(Revisit, Review and Revive)
BTPS R3
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Mathematics 30-1 Learning Outcomes
Topic 1 Trigonometry:
General outcome: develop trigonometric reasoning
Specific outcome 1: Demonstrate an understanding of angles in standard
position, expressed in degrees and radians.
- Sketch, in standard position, an angle (positive or negative) when the measure
is given in degrees.
- Describe the relationship among different systems of angle measurement, with
emphasis on radians and degrees.
- Sketch, in standard position, an angle with a measure of 1 radian.
- Sketch, in standard position, an angle with a measure expressed in the form kπ
radians, where k  Q .
- Express the measure of an angle in radians (exact value or decimal
approximation), given its measure in degrees.
- Express the measure of an angle in degrees, given its measure in radians (exact
value or decimal approximation).
- Determine the measures, in degrees or radians, of all angles in a given domain
that are coterminal with a given angle in standard position.
- Determine the general form of the measures, in degrees or radians, of all angles
that are coterminal with a given angle in standard position.
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Key terms:
terminal arm: The line that tells you where the angle ends.
Reference angle: The acute angle formed between the x axis and the terminal arm.
Principal angle: The smallest positive angle that describes an angle in standard
position. The principal angle may be measured in degrees or radians.
coterminal angles: co means together (like a co-worker), terminal means (where you
end up) so coterminal angles are two angles with different measures that end up at the
same place.
General solution: The general solution is a formula which you can use to find all the
coterminal angles for a known principal angle.
 coterminal   principal  360n, n  I
 coterminal   principal  360n, n  N
 coterminal   principal  2 n, n  I
 coterminal   principal  2 n, n  N
Example 1: For the angle -120° determine the reference angle, a positive coterminal
angle, a negative coterminal angle and the general solution.
Example 2: Sketch the angle
in standard position, determine the reference angle, and
a general solution for all real numbers
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Example 3: Sketch an angle measuring 
2
in standard position. In which quadrant
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does the angle terminate?
Convert between Degrees and Radians
Key ideas: one full rotation is 360° or 2π radians
To convert from radians to degrees multiply by
To convert from degress to radians multiply by
Example 4: Convert 50° to radians
Example 5: Convert
to degrees.
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Determine arc length: Determine the relationship between the radian measure of an
angle in standard position and the length of the arc cut on a circle of radius r, and solve
problems based upon that relationship.
angle in radians
arc length

full rotation angle in radians circumference

a

2 C

a

2 2 r
a

r
NOTE: the angle has to be in Radians, if you are given an angle in degrees, you
need to convert it first.
Example 6: A rotating water sprinkler makes one revolution every 15 s. The water
reaches a distance of 5 m from the sprinkler. What is the arc length of the sector watered
5
when the sprinkler rotates through
? Give your answer as both an exact value and an
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approximate measure to the nearest hundredth.
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Practice:
1.
2. On a circle with a radius of 4.1 cm, an arc of 18.3 cm subtends a central angle of θ to
the nearest tenth of a degree is
A. 255.7°
B.258.9°
C. 256.9°
D. 254.4°
3. An angle, in radians, that is coterminal with 30° is
A.
B.
C.
D.
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Use the following information to answer the next question.
An angle θ, in standard position is shown below.
4. The best estimate of the rotation angle θ is
A. 1.25 radians
B. 3.12 radians
C. 4.01 radians
D. 5.38 radians
NR 1 The two statements that are correct are ________ and ___________
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NR 2: The two statements that are true from the list above are numbered ______ and
______
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5.
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Specific outcome 2: Develop and apply the equation of the unit circle.
Derive the equation of the unit circle from the Pythagorean theorem.
c2  a 2  b2
r x y
2
1
2
2
 Pythagorean Theorem 
2
 x2  y 2
 For the unit circle:
r  1
1  x2  y 2
x2  y 2  1
-
Describe the six trigonometric ratios, using a point P (x, y) that is the
intersection of the terminal arm of an angle and the unit circle.
Coordinate Plane Trigonometry
 For r  0 
 For the unit circle:
y
r
x
cos  
r
y
tan  
x
r
csc  
y
r
sec  
x
x
cot  
y
sin   y
or
y  sin 
cos   x
or
x  cos 
sin  
-
 Equation of a unit circle 


 centred at the origin.

r  1
y
x
1
csc  
y
1
sec  
x
x
cot  
y
tan  
Note: P  x, y  is identical to P  cos  ,sin   .
Generalize the equation of a circle with centre (0, 0) and radius r.
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c2  a 2  b2
 Pythagorean Theorem 
r 2  x2  y 2
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Example 7: if (-3,-4) is a point on a circle. Determine the ratios of the three primary trig
ratios.
Example 8: Is the point (4, 6) on the unit circle?
Example 9: Determine the missing y coordinate if (-1/5, y) is on the unit circle in
quadrant 3.
Practice Questions
NR 3
Use the following information to answer the next question
If the point P (0.2, k), lies on a circle with a radius of 1, the the exact value of k can be
expressed as √ .
NR 4. The value of b, to the nearest hundredth is _________
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6.
Specific Outcome 3: Solve problems, using the six trigonometric ratios for angles
expressed in radians and degrees.
- Determine, with technology, the approximate value of a trigonometric ratio for
any angle with a measure expressed in either degrees or radians.
- Determine, using a unit circle or reference triangle, the exact value of a
trigonometric ratio for angles expressed in degrees that are multiples of 0º, 30º,
45º, 60º or 90º, or for angles expressed in radians that are multiples of 0, π/6,
π/4, π/3, or π/2 , and explain the strategy
- Explain how to determine the exact values of the six trigonometric ratios,
given the coordinates of a point on the terminal arm of an angle in standard
position
- Determine, with or without technology, the measures, in degrees or radians, of
the angles in a specified domain, given the value of a trigonometric ratio.
- Determine the measures of the angles in a specified domain in degrees or
radians, given a point on the terminal arm of an angle in standard position.
- Determine the exact values of the other trigonometric ratios, given the value of
one trigonometric ratio in a specified domain.
- Sketch a diagram to represent a problem that involves trigonometric ratios.
- Solve a problem, using trigonometric ratios.
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Key ideas:
Draw pictures wherever you can. Be careful not to confuse angles and the ratios. Check
the domain to see if you need multiple angles. Ex. If sin A = 0.75, there are two solutions
for the domain of 1 rotation because sine is positive in quadrant 1 and 2.
Calculator TIP
For the TI-83 Plus or the TI-84 Plus
Press MODE
Select RADIAN or DEGREE
QUIT
Press 2ND MODE
Enter the given trigonometric ratio
Press ENTER
For the TI nspire Calculators (note that at the top of the calculator it tells you what
mode you are in)
Press Menu (in graph mode)  settings  graphing angle  select degree or radian
 enter the trig ratio
OR
Press Home button  press 5 settings  Doc settings  change angle to degree or
radian  enter trig ratio
OR
Put your calculator in radian mode and use the degree symbol for angles that are in
degree (found at the same place as the pi button)
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The unit circle: students should either memorize the unit circle or the special triangles
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and CAST rule.
The CAST rule helps you determine in which quadrant each of the primary trig ratios
is positive. The reciprocal trigonometric ratios have the same signs as the primary trig
ratio they are related to.
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Example 10: Determine the approximate for each trigonometric ratio. Give answers to
two decimal places.
a)
b)
c)
d)
e)
sin 5
cos 47
tan 0.94
csc 4.71
sec  15 
f) cot160
Example 11: Determine the exact value for each trigonometric ratio. Explain your
strategy.
 3 
a) sin  
 4 
 7 
b) cos 

 6 
c) tan  120 
d) csc  240 
e) sec  
 11 
f) cot 

 6 
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Example 12: Using the point P  5,  12  , explain how to determine the exact value of
each trigonometric ratio given that P is a point on the terminal arm of an angle  in
standard position.
Example 13: Determine the exact measure of all angles that satisfy sin   
domain 0    2 .
1
in the
2
Example 14: Each point lies on the terminal arm of an angle  in standard position.
Determine  in the specified domain. Round answers to the nearest tenth of a unit.
A  3, 4  ,0    4
B  5,  1 ,  360    360
Example 15: Determine the exact values of the other five trigonometric ratios under the
given conditions.
cos   
2 2
3
,   
3
2
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Example 16: The centre of a clock is placed at the origin of the coordinate plane. If the
tip of the second hand is 5 units from the centre of the clock, sketch its path as it travels
from the vertical position known as 12 o’clock to the horizontal position known as 3
o’clock. Label the point on the quarter-circle where x  2.5 units . Show that this point
is one-third the distance between  0,5  and  5, 0  on the arc of the circle.
Practice:
7.
8.
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NR 5
NR 6
NR 7
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9. If the value of cotA and the value of sec A are both negative, then
A.
B.
C.
D.
10, If
√ , where
, then the exact value of cos θ is
A.
B.
√
√
C.
D.
√
11.
NR 8
NR 9
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Specific outcome 4: Graph and analyze the trigonometric functions sine, cosine and
tangent to solve problems.
- Sketch, with or without technology, the graph of y = sin x, y = cos x or y = tan
x. Determine the characteristics (amplitude, asymptotes, domain, period, range
and zeros) of the graph of y = sin x, y = cos x or y = tan x.
- Determine how varying the value of a affects the graphs of y = a sin x and y =
a cos x.
- Determine how varying the value of d affects the graphs of y = sin x + d and y
= cos x + d.
- Determine how varying the value of c affects the graphs of y = sin (x – c) and y
= cos (x – c).
- Determine how varying the value of b affects the graphs of y = sin bx and y =
cos bx.
- Sketch, without technology, graphs of the form y = a sin b(x − c) + d or y = a
cos b(x − c) + d, using transformations, and explain the strategies.
- Determine the characteristics (amplitude, asymptotes, domain, period, phase
shift, range and zeros) of the graph of a trigonometric function of the form y =
a sin b(x − c) + d or y = a cos b(x − c) + d.
- Determine the values of a, b, c and d for functions of the form y = a sin b(x −
c) + d or y = a cos b(x − c) + d that correspond to a given graph, and write the
equation of the function.
- Determine a trigonometric function that models a situation to solve a problem.
- Explain how the characteristics of the graph of a trigonometric function relate
to the conditions in a problem situation.
- Solve a problem by analyzing the graph of a trigonometric function.
Key ideas:
y  sin 
y  cos 
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y  tan 
Characteristics:
Property
y  sin 
y  cos 
maximum
minimum
amplitude
period
domain
range
y-intercept
xintercept(s)
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y  tan 
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Example 17: Using the language of transformations, describe how to obtain the graph of
each function from the graph of y  sin  or y  cos .
 
 
a) y  3sin  2       6
3 
 
1 
 
b) y  2cos        3
4 
2 
3
c) y  cos  2  60   10
4
d) y   sin  2 x  90   8
Example 18: Graph each function. State the domain and range, the amplitude, the
period, the phase shift, and the x- and y-intercepts of the function.
y  2cos  x  45  3
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 
 
y  4sin  2       1
3 
 
Example 19: Determine an equation in the form y  a sin b   c   d and in the form
y  a cos b   c   d for the following graph.
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Example 20: An electric heater turns on and off on a cyclic basis as it heats the water in
a hot tub. The water temperature, T, in degrees Celsius, varies sinusoidally with time, t,
in minutes. The heater turns on when the temperature of the water reaches 34C and
turns off when the water temperature is 43C . Suppose the water temperature drops to
34C and the heater turns on. After another 30 minutes the heater turns off, and then
after another 30 minutes the heater starts again.
a) Write the equation that expresses temperature as a function of time.
b) Graph the sinusoidal function.
c) Determine the temperature 10 minutes after the heater first turns on
Explain how the characteristics of the graph of the sinusoidal function relate to the
conditions in the problem.
Example 21: The graph of time of sunrise (for latitude 45 degrees) versus months since
the winter solstice is represented by the solid line.
a) Determine the sinusoidal function in the form t  a sin b  M  c   d .
b) Use your sinusoidal function to determine when a 6:30 a.m. sunrise will occur.
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Practice:
12.
13.
14.
15.
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NR 10
NR 11
NR 12
WR 1
16.
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Record you answers for a as NR 13, b for NR 14, and c for NR 15
NR 13:
NR 14:
NR 15
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NR 16
17.
18.
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19.
20.
21.
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22.
23.
24.
25.
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26.
27.
28.
WR 2
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29.
30.
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31.
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Practice Answer Key
1. A
2. A
3. D
4. C
5. C
6. A
7. D
8. D
9. B
10. C
11. B
12. C
13. C
14. C
15. A
16. C
17. B
18. A
19. B
20. C
21. D
22. D
23. A
24. C
25. C
26. A
27. B
28. D
29. C
30. C
31. A
N.R.
1. 1, 3
2. 1, 4
3. 14.1
4. 0.96
5. 0.28
6. 2,1,1,2
7. 7, 12
8. 2.8
9. 539
10. 0.39
11. 20, 0.16, 21
12. 2
13. 812159
14. 17.00
15. 5.66
16. 523
W.R. 1
a)
b)
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