Trigonometry - Weebly · 2020. 3. 15. · Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cos f, sin f) Example 5: The unit circle contains values for cos f and sin f only

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TrigonometryLESSON ONE - Degrees and Radians

Lesson Notes

Example 1: Define each term or phrase and draw a sample angle.

a) Angle in standard position. 'UDZ�D�VWDQGDUG�SRVLWLRQ�DQJOH��ť�

c) Reference angle. )LQG�WKH�UHIHUHQFH�DQJOH�RI�ť� ��

210°×�

180°=��6

b) Positive and negative angles. Draw ť� ��DQG�ť� ��.

d)�&R�WHUPLQDO�DQJOHV��Draw the first positive

FR�WHUPLQDO�DQJOH�RI�� e) Principal angle. )LQG�WKH�SULQFLSDO�DQJOH�RI�ť� ��f)�*HQHUDO�IRUP�RI�FR�WHUPLQDO�DQJOHV��Find the first four

positive�DQG�QHJDWLYH�FR�WHUPLQDO�DQJOHV�RI�ť� ��

Example 2: Three Angle Types: Degrees, Radians, and Revolutions.

a) i. Define degrees. 'UDZ�ť� ��ii. Define radians. 'UDZ�ť� ��UDG��iii. Define revolutions. 'UDZ�ť� ��UHY�

b) Use conversion multipliers to answer the questions and fill in the reference chart.

i.�� BB�UDG��ii.�� BB�UHY��iii.�� BB��iv.�� BB�UHY��v.��UHY� �BB��vi.��UHY� �BBUDG

Conversion Multiplier Reference Chart (Example 2)

degree radian revolution

degree

radian

revolution

c) Contrast the decimal approximation of a radian with the exact value of a radian.

i.�� BBBBB�UDG�(decimal approximation). ii.�� BBBBB�UDG�(exact value).

Example 3: Convert each angle to the requested form. Round all decimals to the nearest hundredth.

a)�FRQYHUW��WR�DQ�DSSUR[LPDWH�UDGLDQ�GHFLPDO��b)�FRQYHUW��WR�DQ�H[DFW�YDOXH�UDGLDQ�c)�FRQYHUW��WR�DQ�H[DFW�YDOXH�UHYROXWLRQ��d)�FRQYHUW�� to degrees. e)�FRQYHUW��to degrees.

f)�ZULWH��as an approximate radian decimal. g)�FRQYHUW��WR�DQ�H[DFW�YDOXH�UHYROXWLRQ�h)�FRQYHUW��UHY�to degrees. i)�FRQYHUW��UHY�to radians.

Example 4: The diagram shows commonly used degrees.

)LQG�H[DFW�YDOXH�UDGLDQV�WKDW�FRUUHVSRQG�WR�HDFK�GHJUHH��When complete, memorize the diagram.

a) Method One: )LQG�DOO�H[DFW�YDOXH�UDGLDQV�XVLQJ�a conversion multiplier.

b) Method Two: Use a shortcut (counting radians).

��

��

��

��

��

��

��

��

��

��

��

��

��

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�� Example 5: Draw each of the following angles in

standard position. State the reference angle.

a)��b)��c)��d)��e)��

Example 6: Draw each of the following angles in

standard position. State the principal and

reference angles.

a)��b)��c) 9 d)��

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Example 10: In addition to the three

primary trigonometric ratios (sinť, cosť,

and tanť), there are three reciprocal

ratios (cscť, secť, and cotť). Given a

triangle with side lengths of x and y,

and a hypotenuse of length r, the six

trigonometric ratios are as follows:

a)�,I�WKH�SRLQW�3��H[LVWV�RQ�WKH�WHUPLQDO�DUP�RI�DQ�DQJOH�ť in standard position, determine the exact

values of all six trigonometric ratios. State the reference angle and the standard position angle.

sinŦ� y

r

cosŦ� x

r

tanŦ� y

x

cscŦ�

secŦ�

cotŦ� x

yr

Ŧ

r

y

r

x

x

y

�cosŦ

�sinŦ

�tanŦ

b)�,I�WKH�SRLQW�3��H[LVWV�RQ�WKH�WHUPLQDO�DUP�RI�DQ�DQJOH�ť in standard position, determine the exact

values of all six trigonometric ratios. State the reference angle and the standard position angle.

Example 11: Determine the sign of each trigonometric ratio in each quadrant.

a) sinť��b)�FRVť��c)�WDQť��d)�FVFť��e)�VHFť��f)�FRWť

g) How do the quadrant signs of the reciprocal trigonometric ratios (cscť, secť, and cotť) compare

to the quadrant signs of the primary trigonometric ratios (sinť, cosť, and tanť)?

Example 12: Given the following conditions, find the quadrant(s) where the angle ť could potentially exist.

a) i. sinť��ii. cosť !��iii. tanť > �

b) i. sinť�!��DQG�cosť !��ii. secť�!��DQG�tanť ��iii. cscť��DQG�cotť !��

c) i. sinť��DQG�cscť� ��ii. cosť� �� and cscť ��iii. secť�!��DQG�tanť ��

Example 13: Given one trigonometric ratio, find the exact values of the other five trigonometric ratios.

State the reference angle and the standard position angle, to the nearest hundredth of a radian.

a) b)

Example 14: Given one trigonometric ratio, find the exact values of the other five trigonometric ratios.

State the reference angle and the standard position angle, to the nearest hundredth of a degree.

b)a)


Lesson Notes��×

��

��

Example 7:�)RU�HDFK�DQJOH��ILQG�DOO�FR�WHUPLQDO�DQJOHV�ZLWKLQ�WKH�VWDWHG�GRPDLQ�a)��'RPDLQ��ť��b)��'RPDLQ��ť��c)��, 'RPDLQ��ť��d) �� 'RPDLQ��ť��

Example 8: For each angle, use estimation to find the principal angle.

a)��b)��c)��d)��

Example 9:�8VH�WKH�JHQHUDO�IRUP�RI�FR�WHUPLQDO�DQJOHV�WR�ILQG�WKH�VSHFLILHG�DQJOH�a)�ť

p� ��)LQG�ť

c��URWDWLRQV�&&��b)�ť

p� �� (Find ť

c��URWDWLRQV�&�

c) ťc� ��)LQG�n and ť

p) d) ť

c� ��(Find n and ť

p)

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Trigonometry

LESSON ONE - Degrees and Radians

Lesson Notes

��×�

��

b)�*LYHQ�WKH�SRLQW�3��0DUN�WULHV�WR�find the reference angle using a sine ratio,

Jordan tries to find it using a cosine ratio,

and Dylan tries to find it using a tangent

ratio. Why does each person get a

different result from their calculator?

Mark’s Calculation

of ť��XVLQJ�VLQH�Jordan’s Calculation

of ť��XVLQJ�FRVLQH�Dylan’s Calculation

of ť��XVLQJ�WDQ�

sinť =3

5

ť� ��

cosť =-4

5

ť� ��

tanť =3

-4

ť� ��

a) When you solve a trigonometric equation in

your calculator, the answer you get for ť can seem

unexpected. Complete the following chart to learn

how the calculator processes your attempt to solve

for ť.

Example 15: Calculating ť with a calculator.If the angle ť�could exist in

either quadrant ___ or ___ ...

The calculator always

picks quadrant

I or II

I or III

I or IV

II or III

II or IV

III or IV

��FP

��FPn

Example 16: The formula for arc length is a� �UŦ, where a is the arc length, Ŧ�is the central angle in radians,

and r is the radius of the circle. The radius and arc length must have the same units.

a)�'HULYH�WKH�IRUPXOD�IRU�DUF�OHQJWK��D� �UŦ��E��H��6ROYH�IRU�WKH�XQNQRZQ�

��FP

r��

a

��FP

��

Ŧ��FP

��FP

��FP ��FP

��FP

9 cm

��

��

��FP

��

Example 17: Area of a circle sector.

a) Derive the formula for the area of a circle sector,r�ť�

$� � . �E��H� Find the area of each shaded region.

b) c) d) e)

b) c) d) e)

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Example 18: The formula for angular speed is , where Ŵ��*UHHN��2PHJD�

÷ťLV�WKH�DQJXODU�VSHHG��÷ť�LV�WKH�FKDQJH�LQ�DQJOH��DQG�÷7�LV�WKH�FKDQJH�LQ�WLPH��Calculate the requested quantity in each scenario. Round all decimals to the

nearest hundredth.

a)�$�ELF\FOH�ZKHHO�PDNHV��FRPSOHWH�UHYROXWLRQV�LQ��PLQXWH��Calculate the angular speed in degrees per second.

b)�$�)HUULV�ZKHHO�URWDWHV��LQ��PLQXWHV��&DOFXODWH�WKH�DQJXODU�VSHHG�LQ�UDGLDQV�SHU�VHFRQG�c)�7KH�PRRQ�RUELWV�(DUWK�RQFH�HYHU\��GD\V��&DOFXODWH�WKH�DQJXODU�VSHHG�LQ�UHYROXWLRQV�SHU�VHFRQG�,I�WKH�DYHUDJH�GLVWDQFH�IURP�WKH�(DUWK�WR�WKH�PRRQ�LV��NP��KRZ�IDU�GRHV�WKH�PRRQ�WUDYHO�in one second?

d)�$�FRROLQJ�IDQ�URWDWHV�ZLWK�DQ�DQJXODU�VSHHG�RI��USP��:KDW�LV�WKH�VSHHG�LQ�USV"e)�$�ELNH�LV�ULGGHQ�DW�D�VSHHG�RI��NP�K��DQG�HDFK�ZKHHO�KDV�D�GLDPHWHU�RI��FP��&DOFXODWH�WKH�angular speed of one of the bicycle wheels and express the answer using revolutions per second.

a) Calculate the angular speed of the satellite.

([SUHVV�WKH�DQVZHU�DV�DQ�H[DFW�YDOXH��LQ�UDGLDQV�VHFRQG�b)�+RZ�PDQ\�NLORPHWUHV�GRHV�WKH�VDWHOOLWH�WUDYHO�LQ�RQH�PLQXWH"�5RXQG�WKH�DQVZHU�WR�WKH�QHDUHVW�KXQGUHGWK�RI�D�NLORPHWUH��

��NP

��NP

Example 19:�$�VDWHOOLWH�RUELWLQJ�(DUWK��NP�DERYH�WKH�VXUIDFH�PDNHV�RQH�FRPSOHWH�UHYROXWLRQ�HYHU\��PLQXWHV��7KH�UDGLXV�RI�(DUWK�LV�DSSUR[LPDWHO\��NP�


Lesson Notes��×

��

��

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TrigonometryLESSON TWO - The Unit Circle

Lesson Notes

a) A circle centered at the origin can be represented by the

relation x2 + y2 = r2, where r is the radius of the circle.

Draw each circle: i. x2 + y2 = 4 ii. x2 + y2 = 49

Example 1: Introduction to Circle Equations.

(cosŦ, sinŦ)

-10

-10

10

10-10

-10

10

10

b) A circle centered at the origin with a radius of 1 has the

equation x2 + y2 = 1. This special circle is called the unit circle.

Draw the unit circle and determine if each point exists on the

circumference of the unit circle: i. (0.6, 0.8) ii. (0.5, 0.5)

1

1

-1

-1

c) Using the equation of the unit circle, x2 + y2 = 1, find the unknown

coordinate of each point. Is there more than one unique answer?

iii. (-1, y)i. ii. , quadrant II. , cosŦ > 0.iv.

Example 2: The following diagram is called the unit circle. Commonly used angles are shown as

radians, and their exact-value coordinates are in brackets. Take a few moments to memorize this

diagram. When you are done, use the blank unit circle on the next page to practice drawing the

unit circle from memory.

b) Draw the unit circle from memory.

a) What are some useful tips to memorize the unit circle?

Example 3: Use the unit circle to find the exact

value of each expression.

a) sin b) cos 180° c) cos ��4

d) sin ��6

e) sin 0 �2

f) cos g) sin ��3

h) cos -120°

�6

Example 4: Use the unit circle to find the exact

value of each expression.

a) cos 420° b) -cos 3� c) sin ��6

d) cos ��3

��4

f) -sin g) cos2 (-840°) h) cos��3

e) sin ��2

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Lesson Notes(cosŦ, sinŦ)

Example 5: The unit circle contains values for cosŦ and

sinŦ only. The other four trigonometric ratios can be

obtained using the identities on the right. Find the exact

values of secŦ and cscŦ in the first quadrant.

tanŦ =sinŦcosŦ

cotŦ =cosŦsinŦ

secŦ =1

cosŦcscŦ =

1

sinŦ

=1

tanŦExample 6: Find the exact values of tanŦ and cotŦ

in the first quadrant.

Example 7: Use symmetry to fill in quadrants II, III,

and IV for secŦ, cscŦ��tanŦ� and cotŦ.

Example 8: Find the exact value of each expression.

a) sec 120° b) sec c) csc �3

d) csc ��4

e) tan ��4

f) -tan g) cot2(270°) h) cot

�6

��6

��2

b) c) d) a)



a) b) c) d)


a) c) d)b)

a) c)

f)

b)

g) h)e)

d) 2

Example 12: Verify each trigonometric statement with a calculator.

Note: Every question in this example has already been seen earlier in the lesson.

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a) What is meant when you are asked to find on the unit circle?

c) How does a half-rotation around the unit circle change the coordinates?

d) How does a quarter-rotation around the unit circle change the coordinates?

��3

b) Find one positive and one negative angle such that 3�ť��

e) What are the coordinates of P(3)? Express coordinates to four decimal places.

,I�ť� ��ILQG�WKH�FRRUGLQDWHV�RI�WKH�SRLQW�KDOIZD\�DURXQG�WKH�XQLW�FLUFOH��6

,I�ť� ��ILQG�WKH�FRRUGLQDWHV�RI�WKH�SRLQW�D�TXDUWHU�UHYROXWLRQ��FORFNZLVH��DURXQG�WKH�XQLW�FLUFOH�

Example 13: Coordinate Relationships on the Unit Circle

b) How is the central angle of the unit circle related to

its corresponding arc length?

a) What is the circumference of the unit circle?

c) If a point on the terminal arm rotates from 3�ť�� (1, 0)

3�ť�� , what is the arc length?

d) What is the arc length from point A to point B

on the unit circle?

Example 14: Circumference and Arc Length of the Unit Circle

to Ŧ

ť

ť

A

B

Diagram for

Example 14 (d).

b) Which trigonometric ratios are restricted to a range of ��\��?

Which trigonometric ratios exist outside that range?

Example 15: Domain and Range of the Unit Circle

a) Is sinŦ = 2 possible? Explain, using the unit circle as a reference.

d) If exists on the unit circle, how can the equation of the

unit circle be used to find sinŦ? How many values for sinŦ are possible?

e) If cosŦ� ��DQG��Ŧ��KRZ�PDQ\�YDOXHV�IRU�sinŦ�are possible?

c) If exists on the unit circle, how can the unit circle

be used to find cosŦ? How many values for cosŦ are possible?

cosť��sinť

cscť��secť

tanť��FRWť

Range Number Line

Chart for Example 15 (b).

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ťA

x

h

ťB

A B

Example 18: From the observation deck of the Calgary Tower, an observer has to

WLOW�WKHLU�KHDG�ťA�GRZQ�WR�VHH�SRLQW�$��DQG�ť

B down to see point B.

a) Show that the height of the observation

x

cotťA - cotť

B

deck is h = .

ground, to the nearest metre?

b) If ťA = , ť

B = , and x = 212.92 m,

how high is the observation deck above the

a) Use the Pythagorean Theorem to prove that the equation of the unit circle is x2 + y2 = 1.

b) Prove that the point where the terminal arm intersects the unit circle, P(Ŧ), has

coordinates of (cosŦ, sinŦ).

c) If the point ť exists on the terminal arm of a unit circle, find the exact values

of the six trigonometric ratios. State the reference angle and standard position angle to the nearest

hundredth of a degree.

Example 16: Unit Circle Proofs

Example 17: In a video game, the graphic of a butterfly needs to be rotated. To make the

butterfly graphic rotate, the programmer uses the equations:

to transform each pixel of the graphic from its original coordinates, (x, y), to its

new coordinates, (x’, y’). Pixels may have positive or negative coordinates.

[·� �[�FRV�ť��\�VLQ�ť

\·� �[�VLQ�ť��\�FRV�ť

b) If a particular pixel has the coordinates (640, 480) after a rotation of , what were the

original coordinates? Round coordinates to the nearest whole pixel.

��4

a) If a particular pixel with coordinates of (250, 100) is rotated by , what are the new

coordinates? Round coordinates to the nearest whole pixel.

�6

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TrigonometryLESSON THREE - Trigonometric Functions I

Lesson Notes

a) b)

0 � �� Ŧ

y

5

-5

0 � ��

20

-20

Ŧ

y

0 ��

40

-40

Ŧ

y

0 �� Ŧ

y

-12

12

c) d)

Example 1: Label all tick marks in the following grids and state the coordinates of each point.

Example 2: Exploring the graph of y = sinť�

d) State the horizontal displacement (phase shift). e) State the vertical displacement.

f) State the ť-intercepts. Write your answer using a general form expression.

g) State the y-intercept. h) State the domain and range.

a) Draw y = sinť��b) State the amplitude. c) State the period.

Example 3: Exploring the graph of y = cosť�




a) Draw y = cosť��b) State the amplitude. c) State the period.

Example 4: Exploring the graph of y = tanť�




a) Draw y = tanť��b) Is it correct to say a tangent graph has an amplitude? c) State the period.

y = asinb(Ŧ - c) + d

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Lesson Notesy = asinb(Ŧ - c) + d

d) y = cosŦ�-1

2

1

2a) y = sinŦ�- 2 b) y = cosŦ�+ 4 c) y = sinŦ�+ 2

1

2-

Example 9: The b Parameter. Graph each function over the stated domain.

Example 10: The b Parameter. Graph each function over the stated domain.

c) y = 2cos Ŧ�- 1 (-2��Ŧ��

b) y = 4cos2Ŧ�+ 6a) y = -sin(3Ŧ� (-2��Ŧ�� (-2��Ŧ��

d) y = sin Ŧ4

3��Ŧ��1

2

a) y = cos2Ŧ b) y = sin3Ŧ��Ŧ�� Ŧ��

c) y = cos Ŧ d) y = sin Ŧ1

3

1

5��Ŧ�� Ŧ��

Example 5: The a Parameter. Graph each function over the domain 0 � ť�� 2�.

a) y = 3sinŦ b) y = -2cosŦ c) y = sinŦ1

2d) y = cosŦ5

2

Example 7: The d Parameter. Graph each function over the domain 0 � ť�� 2�.

c) write a cosine function.a) write a sine function. b) write a sine function. d) write a cosine function.

0

-28

28

��0

-5

5

14

��( )��

0

-1

1

��0

-8

8

��

Example 6: The a Parameter. Determine the trigonometric function corresponding to each graph.

c) write a cosine function.a) write a sine function. b) write a cosine function. d) write a sine function.

Example 8: The d Parameter. Determine the trigonometric function corresponding to each graph.

0

-35

35

��0

-4

4

��0

-32

32

��0

-4

4

��

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Example 12: The c Parameter. Graph each function over the stated domain.

d) (-2��Ŧ��

a) (-4��Ŧ�� b) (-4��Ŧ��

c) (-2��Ŧ��

Example 13: The c Parameter. Graph each function over the stated domain.

d) (-2��Ŧ��

a) 2�Ŧ�2

b) (-2��Ŧ��

c) (-��Ŧ��

0

-6

6

��

-4

4

�� 2

-1

1

-1

1

��

Example 15: a, b, c, & d Parameters. Graph each function over the stated domain.

0

-1

1

��0

-2

2

��0

-4

4

��0

-2

2

��

a) ��Ŧ�� b) ��Ŧ��

d) ��Ŧ��c) ��Ŧ��3-

c) write a cosine function.a) write a cosine function. b) write a sine function. d) write a sine function.

Example 11: The b Parameter. Determine the trigonometric function corresponding to each graph.

c) write a sine function.a) write a cosine function. b) write a sine function. d) write a cosine function.

Example 14: The c Parameter. Determine the trigonometric function corresponding to each graph.

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-2

2

��

-12

12

��

Example 17: Exploring the graph of y = secť�

a) Draw y = secť��b) State the period. c) State the domain and range.

d) Write the general equation of the asymptotes.

e) Given the graph of f(ť) = cosť��GUDZ�\� ��I�ť�1

Ŧ

y

Example 18: Exploring the graph of y = cscť�

a) Draw y = cscť��b) State the period. c) State the domain and range.


e) Given the graph of f(ť) = sinť��GUDZ�\� ��I�ť�1

Ŧ

y

Example 19: Exploring the graph of y = cotť�

a) Draw y = cotť��b) State the period. c) State the domain and range.


e) Given the graph of f(ť) = tanť��GUDZ�\� ��I�ť�1

Ŧ

y

0

-3

3

��

y = secŦ

0

-3

3

y = cscŦ

��

a) b) c) d)

0

-3

3

��

y = secŦ

0

-3

3

��

y = cotŦ

Example 20: Graph each function over the domain 0 � ť�� 2�. State the new domain and range.

2�-2�

2�-2�

2�-2�

3

-3

3

-3

3

-3

Example 16: a, b, c, & d. Determine the trigonometric function corresponding to each graph.

a) write a cosine function. b) write a cosine function.

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TrigonometryLESSON FOUR - Trigonometric Functions II

Lesson Notes

h(t)

t

Example 5: Determine the trigonometric function corresponding to each graph.

0

-10

10

168

a) write a cosine function.

-5

5

168-4

b) write a sine function.

0

10

25

-10

(8, 9)

(16, -3)

c) write a cosine function. d) write a sine function.

0

300

2400

-300

(1425, 150)

(300, -50)

c) If the range of y = 3cosť + d is [-4, k],

determine the values of d and k.b) Find the range of

4.

If the amplitude of each graph is quadrupled, determine the new points of intersection.

e) The graphs of f(ť) and g(ť) intersect at the points

and

Example 6: a) If the transformation g(ť�� I��ť��LV�DSSOLHG�WR�WKH�JUDSK�RI�I�ť�� VLQť��ILQG�WKH�QHZ�UDQJH�

d) State the range of

f(ť) - 2 = msin(2ť) + n.

.

Example 1: Trigonometric Functions of Angles

a) b)

ii. Graph this function using technology. ii. Graph this function using technology.

i. Graph: (0 � ť�< 3�� i. Graph: (0º � ť�< 540º)

Example 2: Trigonometric Functions of Real Numbers.

i. Graph:i. Graph:

c) What are three differences between trigonometric functions of angles and trigonometric

functions of real numbers?

a) b)

ii. Graph this function using technology. ii. Graph this function using technology.

Example 3: Determine the view window for each function and sketch each graph.

b)a)

b)a)

Example 4: Determine the view window for each function and sketch each graph.

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Lesson Notes

h(t)

t

a) If the point lies on the graph of , find the value of a.

b) Find the y-intercept of .

c) The graphs of f(ť) and g(ť) intersect at

the point (m, n). Find the value of f(m) + g(m).

d) The graph of f(ť�� NFRVť�LV�WUDQVIRUPHG�WR�WKH�JUDSK�RI�J�ť�� EFRVť�E\�D�YHUWLFDO�VWUHWFK�about the x-axis.

If the point

of g(ť��state the vertical stretch factor.

exists on the graph

�2

� ��2

��

f(ť)

g(ť)

k

b

Graph for Example 7d

f(ť)

g(ť)

(m, n)

m

n

Graph for Example 7c

Example 8: The graph shows the height of a pendulum bob

as a function of time. One cycle of a pendulum consists of

two swings - a right swing and a left swing.

12 cm

4 cm

8 cm

2 s1 s 3 s 4 s

0 cmt

h(t)

ground level

Graph for Example 8a) Write a function that describes the height of the pendulum

bob as a function of time.

b) If the period of the pendulum is halved, how will this

change the parameters in the function you wrote in part (a)?

c) If the pendulum is lowered so its lowest point is 2 cm

above the ground, how will this change the parameters in

the function you wrote in part (a)?

Example 7:

Example 9: A wind turbine has blades that are 30 m long. An observer notes

that one blade makes 12 complete rotations (clockwise) every minute.

The highest point of the blade during the rotation is 105 m.

a) Using Point A as the starting point of the graph,

draw the height of the blade over two rotations.

b) Write a function that corresponds to the graph.

c) Do we get a different graph if the wind turbine rotates counterclockwise?

A

Example 10: A person is watching a helicopter ascend

from a distance 150 m away from the takeoff point.

a) Write a function, h(Ŧ), that expresses the height as a function of the

angle of elevation. Assume the height of the person is negligible.

b) Draw the graph, using an appropriate domain.

c) Explain how the shape of the graph relates to the motion of the helicopter. Ŧ

h

150 m

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Lesson Notes

h(t)

t

a) Draw the graph for two full oscillations of the mass.

b) Write a sine function that gives the height of the mass

above the ground as a function of time.

c) Calculate the height of the mass after 1.2 seconds.

Round your answer to the nearest hundredth.

d) In one oscillation, how many seconds is the mass lower

than 3.2 m? Round your answer to the nearest hundredth.

Example 11: A mass is attached to a spring 4 m above the ground and allowed to oscillate from its

equilibrium position. The lowest position of the mass is 2.8 m above the ground, and it takes 1 s

for one complete oscillation.

a) Draw the graph for two full rotations of the Ferris wheel.

b) Write a cosine function that gives the height of the rider as a function of time.

c) Calculate the height of the rider after 1.6 rotations of the Ferris wheel.


d) In one rotation, how many seconds is the rider higher than 26 m?


Example 12: A Ferris wheel with a radius of 15 m rotates once every 100 seconds.

Riders board the Ferris wheel using a platform 1 m above the ground.

Example 13: The following table shows the number of daylight hours in Grande Prairie.

December 21

6h, 46m

March 21

12h, 17m

June 21

17h, 49m

September 21

12h, 17m

December 21

6h, 46m

a) Convert each date and time to a number that can be used for graphing.

b) Draw the graph for one complete cycle (winter solstice to winter solstice).

c) Write a cosine function that relates the number of daylight hours, d, to the day number, n.

d) How many daylight hours are there on May 2? Round your answer to the nearest hundredth.

e) In one year, approximately how many days have more than 17 daylight hours?

Round your answer to the nearest day.

December 21 = March 21 = June 21 = September 21 = December 21 =Day Number

Daylight Hours 6h, 46m = 12h, 17m = 17h, 49m = 12h, 17m = 12h, 46m =

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Lesson Notes

h(t)

t

Example 17: Shane is on a Ferris wheel, and his height

Tim, a baseball player, can throw a baseball with a speed of 20 m/s.

If Tim throws a ball directly upwards, the height can be determined

by the equation hball

(t) = -4.905t2 + 20t + 1.

If Tim throws the baseball 15 seconds after the ride begins,

when are Shane and the ball at the same height?

can be described by the equation .

Example 14: The highest tides in the world occur between New Brunswick and

Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and

two high tides. The chart below contains tidal height data that was collected

over a 24-hour period.

a) Convert each time to a decimal hour.

b) Graph the height of the tide for one

full cycle (low tide to low tide).

c) Write a cosine function that relates

the height of the water to the elapsed time.

d) What is the height of the water at 6:09 AM? Round your answer to the

nearest hundredth.

e) For what percentage of the day is the height of the water greater than 11 m?

Round your answer to the nearest tenth.

Bay ofFundy

Bay ofFundy

Time Decimal Hour Height of Water (m)

Low Tide

Low Tide

High Tide

High Tide

2:12 AM

8:12 AM

2:12 PM

8:12 PM

3.48

13.32

3.48

13.32

a) Graph the population of owls and mice over six years.

b) Describe how the graph shows the relationship between owl and mouse populations.

Example 15: A wooded region has an ecosystem that supports both owls and mice.

Owl and mice populations vary over time according to the equations:

Owl population: Mouse population:

where O is the population of owls, M is the population of mice, and t is the time in years.

Example 16: The angle of elevation between the 6:00 position and the 12:00 position

of a historical building’s clock, as measured from an observer

The observer also knows that he is standing 424 m away from the

clock, and his eyes are at the same height as the base of the clock.

The radius of the clock is the same as the length of the minute hand.

If the height of the minute hand’s tip is measured relative to the

bottom of the clock, what is the height of the tip at 5:08,

to the nearest tenth of a metre?

�444

�444

.standing on a hill, is

424 m

Note: Actual tides at

the Bay of Fundy are

6 hours and 13 minutes

apart due to daily

changes in the position

of the moon.

In this example, we will

use 6 hours for simplicity.

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Example 8: Reciprocal Ratios. Find all angles in the domain 0 � ť � 2� that satisfy the given equation.

Write the general solution. Solve equations graphically with intersection points.

f) ťe) ťd) VHFť� ��

TrigonometryLESSON FIVE - Trigonometric Equations

Lesson Notes

a)

Example 1: Primary Ratios. Find all angles in the domain 0 � ť � 2� that satisfy the given equation.

Write the general solution. Solve equations non-graphically using the unit circle.

b) c) 0 d) tan2ť� ��


Write the general solution. Solve equations graphically with intersection points.

a) VLQť b) VLQť� �� c) FRVť d) FRVť� �� e) WDQť f) WDQť� �undefined

Example 3: Primary Ratios. Find all angles in the domain 0°��ť��360° that satisfy the given equation.

Write the general solution. Solve equations non-graphically with a calculator (degree mode).

a) b) c)


Solve equations graphically with ť-intercepts.

b) FRVť� a) VLQť� ��

Solve - 0 � ť � 2�Example 5: Primary Ratios.

6ROYH�VLQť� ��ť�Ţ�5Example 6: Primary Ratios.

Example 7: Reciprocal Ratios. Find all angles in the domain 0 � ť � 2� that satisfy the given equation.

Write the general solution. Solve equations non-graphically using the unit circle.

a) c)b)

a) ť b) ť c) ť

b)a) c)

Example 9: Reciprocal Ratios. Find all angles in the domain 0°��ť��360° that satisfy the

given equation. Write the general solution. Solve non-graphically with a calculator (degree mode).

a)�QRQ�JUDSKLFDOO\��XVLQJ�WKH�cos�� feature of a calculator.

b)�QRQ�JUDSKLFDOO\��using the unit circle.

c)�JUDSKLFDOO\��XVLQJSRLQW�V��RI�LQWHUVHFWLRQ�

d)�JUDSKLFDOO\��using ť�LQWHUFHSWV�



c)�JUDSKLFDOO\��XVLQJSRLQW�V��RI�LQWHUVHFWLRQ�


a) θ b) θ

Example 10: Reciprocal Ratios. Find all angles in the domain 0 � ť � 2� that satisfy the

given equation. Write the general solution. Solve equations graphically with ť-intercepts.

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TrigonometryLESSON FIVE - Trigonometric Equations

Lesson Notes

Solve cscť� ��0 � ť � 2�

Solve secť� �� 0°��ť��360°

Example 11: Reciprocal Ratios.



c)�JUDSKLFDOO\��XVLQJ�SRLQW�V��RI�LQWHUVHFWLRQ�


Example 12: Reciprocal Ratios.

Example 13: First-Degree Trigonometric Equations. Find all angles in the domain 0 � ť � 2� that

satisfy the given equation. Write the general solution.

d)��VHFť�� VHFť��c)��WDQť�� a)�FRVť�� b) ť

Example 14: First-Degree Trigonometric Equations. Find all angles in the domain 0 � ť � 2� that


b)��VLQť� ��VLQťa)��VLQťFRVť� �FRVť c)�VLQťWDQť� �VLQť d)�WDQť��FRVťWDQť� ��

Example 15: Second-Degree Trigonometric Equations. Find all angles in the domain 0 � ť � 2� that


a) sin2ť� �� c) 2cos2ť� �FRVť d) tan4ť��WDQ2ť� ��b) 4cos2ť��

Example 16: Second-Degree Trigonometric Equations. Find all angles in the domain 0 � ť � 2� that


a) 2sin2ť��VLQť�� b) csc2ť��FVFť�� c) 2sin3ť��VLQ2ť��VLQť� ��

Example 17: Double and Triple Angles.�6ROYH�HDFK�HTXDWLRQ��L��JUDSKLFDOO\��DQG��LL��QRQ�JUDSKLFDOO\�

a) ť 0 � ť � 2� b) ť 0 � ť � 2�

Example 18: Half and Quarter Angles.�6ROYH�HDFK�HTXDWLRQ��L��JUDSKLFDOO\��DQG��LL��QRQ�JUDSKLFDOO\�

a) ť 0 � ť � 4� b) ť 0 � ť � 8��

Example 19:�,W�WDNHV�WKH�PRRQ�DSSUR[LPDWHO\��GD\V�WR�JR�WKURXJK�DOO�RI�LWV�SKDVHV�



c)�JUDSKLFDOO\��XVLQJ�SRLQW�V��RI�LQWHUVHFWLRQ�


Example 20: $�URWDWLQJ�VSULQNOHU�LV�SRVLWLRQHG��P�DZD\�IURP�WKH�ZDOO�RI�D�KRXVH��7KH�ZDOO�LV��P�ORQJ��$V�WKH�VSULQNOHU�URWDWHV��WKH�VWUHDP�RI�ZDWHU�VSODVKHV�WKH�KRXVH�d�PHWHUV�IURP�SRLQW�3�

a)�:ULWH�D�WDQJHQW�IXQFWLRQ��G�ť��WKDW�H[SUHVVHV�WKH�GLVWDQFH�ZKHUH�WKH�ZDWHU�VSODVKHV�WKH�ZDOO�DV�D�IXQFWLRQ�RI�WKH�URWDWLRQ�DQJOH�ť.

b)�*UDSK�WKH�IXQFWLRQ�IRU�RQH�FRPSOHWH�URWDWLRQ�RI�WKH�VSULQNOHU��'UDZ�RQO\�WKH�SRUWLRQ�RI�WKH�JUDSK�WKDW�DFWXDOO\�FRUUHVSRQGV�WR�WKH�ZDOO�EHLQJ�VSODVKHG�c)�,I�WKH�ZDWHU�VSODVKHV�WKH�ZDOO��P�QRUWK�RI�SRLQW�3��ZKDW�LV�WKH�DQJOH�of rotation (in degrees)?

a)�:ULWH�D�IXQFWLRQ��3�W��WKDW�H[SUHVVHV�WKH�YLVLEOH�SHUFHQWDJH�RI�WKH�PRRQ�DV�D�IXQFWLRQ�RI�WLPH��'UDZ�WKH�JUDSK��b)�,Q�RQH�F\FOH��IRU�KRZ�PDQ\�GD\V�LV��RU�PRUH�RI�WKH�PRRQ·V�VXUIDFH�YLVLEOH"

3Ŧ

d

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TrigonometryLESSON SIX - Trigonometric Identities I

Lesson Notes

Example 3: Reciprocal Identities. Prove that each trigonometric statement is an identity.

State the non-permissible values of x so the identity is true.

a) b)

Example 4: Reciprocal Identities. Prove that each trigonometric statement is an identity.


a) b)

Example 5: Pythagorean Identities. Prove that each trigonometric statement is an identity.


Example 6: Pythagorean Identities. Prove that each trigonometric statement is an identity.


Example 1: Understanding Trigonometric Identities.

a) Why are trigonometric identities considered to be a special type of trigonometric equation?

b) Which of the following trigonometric equations are also trigonometric identities?

i. ii. iii. iv. v.

a) Using the definition of the unit circle, derive the identity sin2x + cos2x = 1.

Why is sin2x + cos2x = 1 called a Pythagorean Identity?

b) Verify that sin2x + cos2x = 1 is an identity using (i) x = and (ii) x = .

c) Verify that sin2x + cos2x = 1 is an identity using a graphing calculator to draw the graph.

Example 2: The Pythagorean Identities.

d) Using the identity sin2x + cos2x = 1, derive 1 + cot2x = csc2x and tan2x + 1 = sec2x.

e) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities for x = .

f) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities graphically.

a) b)

c) d)

c) d)

a) b)

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TrigonometryLESSON SIX- Trigonometric Identities I

Lesson Notes

Example 7: Common Denominator Proofs. Prove that each trigonometric statement is an identity.


Example 8: Common Denominator Proofs. Prove that each trigonometric statement is an identity.


Example 12: Exploring the proof of

d)c)

a) b)

c) d)

a) b)

Example 9: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.

a) b)

c) d)


a)

c) d)

b)

a) b)

c) d)


a) Prove algebraically that .

b) Verify that for �3

.

c) State the non-permissible values for .

d) Show graphically that . Are the graphs exactly the same?

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a) b)

c) d)

TrigonometryLESSON SIX - Trigonometric Identities I

Lesson Notes


a) Prove algebraically that .

b) Verify that for �3

.

c) State the non-permissible values for .

Example 13: Exploring the proof of .

Example 14: Exploring the proof of


a) Prove algebraically that

b) Verify that for�2

.

c) State the the non-permissible values for .

Example 15: Equations with Identites.�6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

a) b) c) d)

Example 16: Equations with Identites. 6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

a)

c)

b)

d)

Example 17: Equations with Identites. 6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

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TrigonometryLESSON SIX- Trigonometric Identities I

Lesson Notes

Example 18: Use the Pythagorean identities to find the indicated value and

draw the corresponding triangle.

b) If the value of , find the value of secA within the same domain.

c)�,I��ILQG�WKH�H[DFW�YDOXH�RI�VLQť� 7cosť = , and cotť < 0,

7

a) If the value of find the value of cosx within the same domain.

Example 19: Trigonometric Substitution.

3

Ŧ

a) Using the triangle to the right, show that

b

aHint: Use the triangle to find a trigonometric expression equivalent to b.

can be expressed as .

b) Using the triangle to the right, show that

Ŧ

4

aHint: Use the triangle to find a trigonometric expression equivalent to a.

can be expressed as .

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Example 1: Evaluate each trigonometric sum or difference.

TrigonometryLESSON SEVEN - Trigonometric Identities II

Lesson Notes

a) b) d) e) f)c)

Example 2: Write each expression as a single trigonometric ratio.

a) c)b)


a) b) c) d) Given the exact values of cosine and sine for 15°,

fill in the blanks for the other angles.

Example 4: Find the exact value

of each expression.

b) c)a)

Example 5: Double-angle identities.

a) Prove the double-angle sine identity, sin2x = 2sinxcosx.

b) Prove the double-angle cosine identity, cos2x = cos2x - sin2x.

c) The double-angle cosine identity, cos2x = cos2x - sin2x, can be

expressed as cos2x = 1 - 2sin2x or cos2x = 2cos2x - 1. Derive each identity.

d) Derive the double-angle tan identity, .

a) Evaluate each of the following expressions using a double-angle identity.

b) Express each of the following expressions using a double-angle identity.

c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.

Example 6: Double-angle identities.

i.

i.

i.

ii.

ii.

ii.

iii.

iii.

iii.

iv.

iv.

Example 3d

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TrigonometryLESSON SEVEN- Trigonometric Identities II

Lesson Notes

Note: Variable restrictions may be

ignored for the proofs in this lesson.Example 7: Prove each trigonometric identity.

b)a)

c) d)

Example 8: Prove each trigonometric identity.

a) b)

c)


a) b)

c) d)

a)

c) d)

b)



a) b)

d)c)


d)

c) d)

a) b)

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TrigonometryLESSON SEVEN - Trigonometric Identities II

Lesson Notes

b)

d)c)


a)

a)

c) d)

Example 14:�6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

b)

a) b)

c) d)

Example 15:�6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

a) b)

c) d)

Solve for x. Round your answer to the nearest tenth.

a)

c)

b)

Example 16: 6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

d)

Example 17: 6ROYH�HDFK�WULJRQRPHWULF�HTXDWLRQ�RYHU�WKH�GRPDLQ��[��

b) If A = 32° and B = 89°,

what is the value of C? a) Show that

Example 18: Trigonometric identities and geometry.

A

C

B

Diagram forExample 18

Example 19: Trigonometric identities and geometry.

57

176

153

104

A

B

x

Diagram forExample 19

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Trigonometry - Weebly · 2020. 3. 15. · Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cos f, sin f) Example 5: The unit circle contains values for cos f and sin f only