Unit 3 Trigonometry Note Package. Name:

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1 MAT40S Unit 3 Trigonometry Mr. Morris Lesson Unit 3 Trigonometry Note Package Homework 1: Converting and Arc Extra Practice Sheet 1 Length 2: Unit Circle and Angles Extra Practice Sheet 2 3: Determining Exact Extra Practice Sheet 3 Values 4: Graphing Extra Practice Sheet 4 Trigonometric Functions 5: Graphing Reciprocal Nil and Tan Functions 6: Word Problems Extra Practice Sheet 6 Name:

2 MAT40S Trigonometry Page 1 of 76 Objective: Lesson 1 Up until now you have been using degrees as measurements for angles makes a full circle 180 makes half a circle This year we will be learning about radians and using them in combination with degrees to describe angles Radians are useful to use since they are dealt with in terms of π and can be used as exact values instead of approximations we have with degrees. Half a circle is π radians. A full circle is 2π rads No units are associated with radians since it is ratio. But they are also called rads In some instances we will need to convert from degree to radians and vice versa

3 MAT40S Trigonometry Page 2 of 76 Radians Degrees: Muliply by 180 π Degrees Radians: Multiply by π 180 Example 1: Convert 201 to a radian decimal Example 3: Convert 3.24 radians to degrees Example 4: Convert rad to degrees

4 MAT40S Trigonometry Page 3 of 76 Arc Length is another important aspect of circle geometry that you actually already know. Way back when you learned the circumference of a circle is 2πr you were actually using the formula for arc length s θr s = arc length θ = angle measurement (in radians) r = radius of circle Example 7: Determine the length of the arc if a radius of 4m sweeps through an angle of 185.6

5 MAT40S Trigonometry Page 4 of 76 Example 8: Determine the angle, in degrees if a radius of 2.3m cuts an arc of 6.1m

6 MAT40S Trigonometry Page 5 of 76 Example 10: Given the length of the arc is 16cm and the central angel is 80 a) Determine the length of the radius b) Determine the area of the circle c) Determine the area of the sector of the circle

7 MAT40S Trigonometry Page 6 of 76 End of Lesson 1 Practice sheets: Sheet 1

8 MAT40S Trigonometry Page 7 of 76 Objective: Lesson 2 Recall from last year the CAST rule when determining the trigonometric functions that are positive in specific quadrants in the unit circle

9 MAT40S Trigonometry Page 8 of 76 Also recall putting reference angles in each of the 4 quadrants following this formula Quadrant 1 θ Quadrant 2 θ Quadrant 3 θ Quadrant 4 θ = Principal angle = 180 Principal angle = 180 Principal angle = 360 Principal angle In Grade 12.. Quadrant 1 θ Quadrant 2 θ Quadrant 3 θ Quadrant 4 θ = Principal angle = π Principal angle = π Principal angle = 2π Principal angle

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11 MAT40S Trigonometry Page 10 of 76 Recall some terminology from grade 11 that we will be coming back to in grade 12. Principal angle Any angle that has it s terminal arm ending in one of the 4 quadrant from 0 θ 360 Terminal arm The line that comes out of the origin and sweeps some arc making an angle with the initial arm (usually the x axis) Standard position Any angle that has its initial arm along the x axis and has a terminal arm rotating counterclockwise from it

12 MAT40S Trigonometry Page 11 of 76 We can also draw angles in the clockwise direction. These will be negative angles Reference angle the angle formed between the terminal arm and the closest x axis. This will also tell you what family of angles your standard angle belongs to. 300

13 MAT40S Trigonometry Page 12 of 76 Co terminal angle Any terminal arm rotated 360 in either direction negative

14 MAT40S Trigonometry Page 13 of 76 There is a general solution in finding all possible co terminal angle over all real numbers. It can be done in both degrees and radians All possible co terminal angles in degrees: θ 360 k k I All possible co terminal angles in radians: θ 2πk k I

15 MAT40S Trigonometry Page 14 of 76 Example 1: Determine both a positive and negative co terminal angle for. Sketch the solutions

16 MAT40S Trigonometry Page 15 of 76 Example 4: Find the general solution for all coterminal angles for 101 and state the value of the 14 th positive co terminal angle. Example 5: Sketch the following angles in standard position. State the reference angle. a) θ 2

17 MAT40S Trigonometry Page 16 of 76 b) θ c) θ 900

18 MAT40S Trigonometry Page 17 of 76 Example 5: For each angle, find all co terminal angles with the stated domain a), 4π,4π Example 6: State the following answers over [0,2π] (determine the positive co terminal angles with one revolution) a) b) 480 c)

19 MAT40S Trigonometry Page 18 of 76 Recall from grade 11 the new equations that were used for the trigonometric functions based on the unit circle sinθ opposite hypotenuse cosθ adjacent hypotenuse tanθ opposite adjacent A couple of new things will be added this year Reciprocal trig functions are a new addition that include the reciprocal of the standard 3 trig equations cscθ 1 sinθ secθ 1 cosθ cotθ 1 tanθ hypotenuse opposite hypotenus adjacent adjacent opposite

20 MAT40S Trigonometry Page 19 of 76 These 3 new trig functions are all based on the original formulas you have learned previously, just reciprocated. It is important to remember that the hypotenuse or r is always positive on the unite circle The formula for a circle is x y r Since the unit circle had a radius of 1 the equation for the unite circle becomes x 2 y 2 1

21 MAT40S Trigonometry Page 20 of 76 Example 8: Given the following conditions, determine the quadrant(s) where the angle could exist

22 MAT40S Trigonometry Page 21 of 76 Example 9: If the point P: 5,12 exists on the terminal arm of an angle θ in standard position, a) Determine the exact value of all six trigonometric ratios

23 MAT40S Trigonometry Page 22 of 76 b) State the reference angle and the standard angle position

24 MAT40S Trigonometry Page 23 of 76 Example 10: Determine if the point 2, on the unit circle. Justify your answer. Example 11: Determine the unknown coordinate of each point, given that it is on the unit circle. Is there more than one unique answer? a) x, in QII

25 MAT40S Trigonometry Page 24 of 76 Example 12: Given a point with a coordinate 5,12 determine the corresponding point on the unit circle.

26 MAT40S Trigonometry Page 25 of 76 Example 13: If cosθ, determine sinθ, given that θ is not in quadrant 2

27 MAT40S Trigonometry Page 26 of 76 Example 14: Determine cotθ over,π given sinθ End of Lesson 2 Practice sheets: Sheet 2

28 MAT40S Trigonometry Page 27 of 76 Objective: Lesson 3 cos 105 sin 306 cot 495 Use your calculator to solve for the following expressions. Be sure to use your rad function on your calculator when the angle is given in radians sin tan cos 0.97

29 MAT40S Trigonometry Page 28 of 76 In grade 12 you must know the exact values for the unit circle (both radians and degrees). Just as in grade 11 find out what the principal angle is, then use the unit circle and CAST rule to determine the exact values All principal values are from the first quadrant so memorizing those exact values are a must

30 MAT40S Trigonometry Page 29 of 76 Example 1: Use the unit circle to determine the exact value of each trigonometric ratio a) sin b) cos 180 c) sin d) sin e) cos 120

31 MAT40S Trigonometry Page 30 of 76 Example 2: Determine the exact value of the following a) cos 420 b) cos 3π c) cos d) sin e) cos 840

32 MAT40S Trigonometry Page 31 of 76 Example 3: Determine the exact value of the following a) sec 120 b) csc c) tan d) cot 270 e) cot

33 MAT40S Trigonometry Page 32 of 76 Example 4: Determine the exact values of each trigonometric expression a) sin cos b) cos sin c) 2sin 2cos

34 MAT40S Trigonometry Page 33 of 76 d) sin cos 330 tan sin

35 MAT40S Trigonometry Page 34 of 76 When solving for angles instead of values you have to identify if you are dealing with an exact value o 0, 1,, and values for sin and cos o 0, 1, for tan are exact and 3 are exact values If you aren t dealing with one, then you ll have to use your calculator to figure out the angle Remember to cover all possible angles since more than one solution may apply Example 6: Given cos θ, determine the exact value of θ over the interval 0,2π

36 MAT40S Trigonometry Page 35 of 76 Example 8: Determine θ over the interval 0, 2π given csc θ 2

37 MAT40S Trigonometry Page 36 of 76 If we are not given the interval over which to solve for we need to provide a general solution to the problem. This can be asked in a variety of ways to you, but the most common way of identifying if you need a general solution is if the questions asks Determine the general solution Given the domain is all real numbers Given θ R Given θ over, Example 11: Solve for θ, given sin θ a) Over 0, 4π b) Over,

38 MAT40S Trigonometry Page 37 of 76 Example 13: Find the general solution to tan θ 1**** End of Lesson 3 Practice sheets: Sheet 3

39 MAT40S Trigonometry Page 38 of 76 Objective: Lesson 4 Our next task with trigonometric is to graph and analyze plots that use the primary three ratios. A couple of terms and definitions should be covered initially: Period: How often a trigonometric function repeats itself Amplitude: The absolute vertical distance a trigonometric function can go from its mid line More definitions will come up as we go along through this outcome Really important to know the exact values of our trigonometric ratios so we can accurately portray the functions on a graph

40 MAT40S Trigonometry Page 39 of 76 Create a table of values using your unit circle to graph y sinθ for 0 θ 2π Determine the following for all values of y sinθ a) Domain/Range b) θ intercept (x intercept) c) Amplitude d) Period

41 MAT40S Trigonometry Page 40 of 76 Create a table of values using your unit circle to graph y cosθ for 0 θ 2π Determine the following for all values of y sinθ a) Domain/Range b) θ intercept (x intercept) c) Amplitude d) Period

42 MAT40S Trigonometry Page 41 of 76 Now that we have the basic sine and cosine graph we can start apply our transformations to them. Vertical stretches/compressions y asin θ or acos θ All y values are multiplied by the value of a A negative sign is a reflection over the x axis The value of a represents the amplitude of the graph Later on we will have a midline that is not on y 0 so it is important we also learn a formula for amplitude that will become useful later on Amplitude max min 2

43 MAT40S Trigonometry Page 42 of 76 Where max is the maximum y value of the graph and min is the minimum y value of the graph Example 1: Determine the amplitude of the graph of each function: a) y sin x b) y 4cosθ

44 MAT40S Trigonometry Page 43 of 76 Example 2: Sketch the following functions: a) y sin θ for θ R

45 MAT40S Trigonometry Page 44 of 76 Vertical Translations y sinx d or y cosx d Add d to all y values Will move the entire graph up or down Gives the equation for the midline/center line/median/average/ Can use a formula to find it with y Example 3: Sketch the graph of y sinx 3 over 0, 360

46 MAT40S Trigonometry Page 45 of 76 Give the equation of the trigonometric graph below

47 MAT40S Trigonometry Page 46 of 76 Combining vertical stretches and shifts y asinθ d Apply the amplitude stretch or compression first Apply the vertical translation/shift last Example 4: Sketch the graph of y 2cosθ 1 over,2π

48 MAT40S Trigonometry Page 47 of 76 Give the equation of the line below

49 MAT40S Trigonometry Page 48 of 76 Horizontal stretches/compressions y sinbx or y cosbx Divide all x values by b (remember b acts on the graph as its reciprocal) The b value effects the period of the graph y cosx y cos2x y cos x

50 MAT40S Trigonometry Page 49 of 76 Example 5: What is the period of these graphs?

51 MAT40S Trigonometry Page 50 of 76 Relating the b value to what the actual period is uses a specific formula only for sin x and cos x. tan x will be discussed later or or It is absolutely critical that you know that the b value is not the period alone. Example 6: Sketch the graph of one full cycle (period) y cos 1 2 x

52 MAT40S Trigonometry Page 51 of 76 Determine the equation for this graph

53 MAT40S Trigonometry Page 52 of 76 Horizontal translations (also called phase shift) y sin x c or y cos x c The c value with move the graph to the left or to the right Remember to do whatever the opposite operation is Example 7: Sketch the graphs use mapping notation if needed y sin θ for θ over all real numbers

54 MAT40S Trigonometry Page 53 of 76 Find out the equation of this graph*****

55 MAT40S Trigonometry Page 54 of 76 Combining horizontal stretches with shifts Always apple the stretch first (affects the period) FACTOR IF NEEDED Phase shift (horizontal shift) goes last. Remember that the b value is not the period but rather affects the period using the formula period 2π b Example 9: Sketch the following y sin 2θ over 0, 2π y cos 1 2 θ π 4

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57 MAT40S Trigonometry Page 56 of 76 Combining all transformations at once These questions can be challenging, but follow the steps below and the questions will become easier 1) Determine the new period (based on b value) 2) Determine the midline (based on d value) 3) Determine the maximum and minimum (based on a value) 4) Sketch the out the shape lightly (based on cosine or sine) 5) Determine the phase shift (based on c value) 6) Draw the graph completely taking into account the domain (if provided) 7) Or just use mapping notation and do everything at once.

58 MAT40S Trigonometry Page 57 of 76 Example 10 Sketch the following over 0, 2π y 2cos x π 2 1

59 MAT40S Trigonometry Page 58 of 76 y 2cos θ 3 Lesson 5 End of Lesson 4 Practice sheets: Sheet 4

60 MAT40S Trigonometry Page 59 of 76 Objective: Lesson 5 Create a table of values, using your unit circle, to graph y tanθ for 0 θ 2π

61 MAT40S Trigonometry Page 60 of 76 Determine the following for all values of y tanθ a) Domain/range b) θ intercept(s): c) Amplitude: d) Period: e) Equation of asymptote(s) over R

62 MAT40S Trigonometry Page 61 of 76 y cotθ for 0 θ 2π******

63 MAT40S Trigonometry Page 62 of 76 Determine the following for all values of y cotθ a) Domain/Range b) θ intercept(s): c) Amplitude: d) Period: e) Equation of asymptote(s) over R

64 MAT40S Trigonometry Page 63 of 76 y cscθ for 0 θ 2π

65 MAT40S Trigonometry Page 64 of 76 Determine the following for all values of y cscθ a) Domain/range b) θ intercept(s): c) Amplitude: d) Period: e) Equation of asymptote(s) over R

66 MAT40S Trigonometry Page 65 of 76 y secθ for 0 θ 2π

67 MAT40S Trigonometry Page 66 of 76 Determine the following for all values of a) Domain/range y secθ b) θ intercept(s): c) Amplitude: d) Period: e) Equation of asymptote(s) over R Lesson 6 End of Lesson 5 Practice sheets: Nil

68 MAT40S Trigonometry Page 67 of 76 Objective: Lesson 6 Given a word problem, or a graph to analyze, you will have to determine the trigonometric equation. This means finding a, b, c, and d values in the y asin b x c d or y acos b x c d Sometimes a question will specify if you are to use sine or cosine function but usually it is up to you to decide what you what to use Mapping notation can always be used to help you out. All of the rules that we have covered with transformations still apply here.we just have different functions now!

69 MAT40S Trigonometry Page 68 of 76 Recall the following formulas to help you 1) Finding a (amplitude) amplitude max min 2 2) Finding b (related to the period) b 2π period or b 360 period period 2π or period 360 b period 3) Finding d (the midline/average/etc) min max d 2 4) Finding c (the phase shift) i. If creating a cosine equation find the maximum and determine the horizontal translation from the y axis ii. If creating a sine equation find the closest point where the graph intersects the midline and determine the horizontal translation from the y axis

70 MAT40S Trigonometry Page 69 of 76 Remember that these c values will be different depending on if you re going with cosine or with a sine function! Also remember that whichever way you determine the horizontal shift to be it is the opposite that goes inside the bracket with x! With that, you are ready to identify and analyze trigonometric functions!

71 MAT40S Trigonometry Page 70 of 76 Example 1: Determine the trigonometric function corresponding to each graph

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73 MAT40S Trigonometry Page 72 of 76 Example 2: 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 a) 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 function you wrote in part a)?

74 MAT40S Trigonometry Page 73 of 76 Example 3: A wind turbine has blades that are 30m long. An observer notes that one blade makes 12 complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 105m a) Using Point A as the starting point of the graph. Draw the height of the blade over two rotations b) Write an equation that corresponds to the graph. c) If the blades rotates counterclockwise, do we get a different function?

75 MAT40S Trigonometry Page 74 of 76 Example 5: A Ferris wheel with a radius of 15m rotates once every 100 seconds. Riders board the Ferris wheel using a platform 1m above the ground. 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.

76 MAT40S Trigonometry Page 75 of 76 Example 7: Given the following sinusoidal equation P t 3000 sin π t Determine the maximum value of P t and a value of t at which this maximum occurs.

77 MAT40S Trigonometry Page 76 of 76 End of Lesson 6 Practice sheets: Sheet 6