Mathematics UNIT FIVE Trigonometry II. Unit. Student Workbook. Lesson 1: Trigonometric Equations Approximate Completion Time: 4 Days

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1 Mathematics 0- Student Workbook Unit 5 Lesson : Trigonometric Equations Approximate Completion Time: 4 Days Lesson : Trigonometric Identities I Approximate Completion Time: 4 Days Lesson : Trigonometric Identities II Approximate Completion Time: 4 Days UNIT FIVE Trigonometry II

2 Mathematics 0- Unit 5 Student Workbook Complete this workbook by watching the videos on Work neatly and use proper mathematical form in your notes. UNIT FIVE Trigonometry II

3 LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with the unit circle. a) b) c) 0 d) tan θ =

4 LESSON FIVE - Trigonometric Equations a) Example sinθ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) sinθ = - Primary Ratios Solving equations graphically with intersection points - - c) cosθ d) cosθ = e) tanθ f) tanθ = undefined

5 LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with a calculator. (degree mode) a) b) c)

6 LESSON FIVE - Trigonometric Equations Example 4 a) sinθ = Find all angles in the domain 0 θ that satisfy the given equation. Intersection Point(s) of Original Equation Primary Ratios Solving equations graphically with θ-intercepts. θ-intercepts b) cosθ = Intersection Point(s) of Original Equation θ-intercepts

7 LESSON FIVE - Trigonometric Equations Example 5 Solve a) non-graphically, using the cos - feature of a calculator. - 0 θ Primary Ratios Equations with b) non-graphically, using primary trig ratios the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

8 LESSON FIVE - Trigonometric Equations Example 6 a) non-graphically, using the sin - feature of a calculator. Solve sinθ = -0.0 θ ε R Primary Ratios Equations with primary trig ratios b) non-graphically, using the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

9 LESSON FIVE - Trigonometric Equations Example 7 Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Reciprocal Ratios Solving equations with the unit circle. a) b) c)

10 LESSON FIVE - Trigonometric Equations Example 8 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) θ Reciprocal Ratios Solving equations graphically with intersection points c) θ d) secθ = e) θ f) θ

11 LESSON FIVE - Trigonometric Equations Example 9 Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution Reciprocal Ratios Solving equations with a calculator. (degree mode) a) b) c)

12 LESSON FIVE - Trigonometric Equations Example 0 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Intersection Point(s) of Original Equation Reciprocal Ratios Solving equations graphically with θ-intercepts. θ-intercepts b) θ Intersection Point(s) of Original Equation θ-intercepts

13 LESSON FIVE - Trigonometric Equations Example a) non-graphically, using the sin - feature of a calculator. Solve cscθ = - 0 θ Reciprocal Ratios b) non-graphically, using the unit circle. Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

14 LESSON FIVE - Trigonometric Equations Example a) non-graphically, using the cos - feature of a calculator. Solve secθ = θ 60 b) non-graphically, using the unit circle. Reciprocal Ratios Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts

15 LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. a) cosθ - = 0 b) θ First-Degree Trigonometric Equations c) tanθ - 5 = 0 d) 4secθ + = secθ +

16 LESSON FIVE - Trigonometric Equations Example 4 Find all angles in the domain 0 θ that satisfy the given equation. a) sinθcosθ = cosθ b) 7sinθ = 4sinθ First-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) sinθtanθ = sinθ d) tanθ + cosθtanθ = 0 Check the solution graphically. Check the solution graphically

17 LESSON FIVE - Trigonometric Equations Example 5 Find all angles in the domain 0 θ that satisfy the given equation. a) sin θ = b) 4cos θ - = 0 Second-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) cos θ = cosθ d) tan 4 θ - tan θ = 0 Check the solution graphically. Check the solution graphically

18 LESSON FIVE - Trigonometric Equations Example 6 a) sin θ - sinθ - = 0 Find all angles in the domain 0 θ that satisfy the given equation. Second-Degree Trigonometric Equations Check the solution graphically b) csc θ - cscθ + = 0 Check the solution graphically c) sin θ - 5sin θ + sinθ = 0 Check the solution graphically

19 LESSON FIVE - Trigonometric Equations Example 7 Solve each trigonometric equation. Double and Triple Angles a) θ 0 θ i) graphically: ii) non-graphically: - b) θ 0 θ i) graphically: ii) non-graphically: -

20 LESSON FIVE - Trigonometric Equations Example 8 Solve each trigonometric equation. Half and Quarter Angles a) θ 0 θ 4 i) graphically: ii) non-graphically: 4 - b) θ - 0 θ 8 i) graphically: ii) non-graphically:

21 LESSON FIVE - Trigonometric Equations Example 9 It takes the moon approximately 8 days to go through all of its phases. New Moon First Quarter Full Moon Last Quarter New Moon a) Write a function, P(t), that expresses the visible percentage of the moon as a function of time. Draw the graph. Visible % t b) In one cycle, for how many days is 60% or more of the moon s surface visible?

22 LESSON FIVE - Trigonometric Equations Example 0 Rotating Sprinkler N A rotating sprinkler is positioned 4 m away from the wall of a house. The wall is 8 m long. As the sprinkler rotates, the stream of water splashes the house d meters from point P. Note: North of point P is a positive distance, and south of point P is a negative distance. a) Write a tangent function, d(θ), that expresses the distance where the water splashes the wall as a function of the rotation angle θ. W S E θ P d b) Graph the function for one complete rotation of the sprinkler. Draw only the portion of the graph that actually corresponds to the wall being splashed. 8 4 d -4 θ -8 c) If the water splashes the wall.0 m north of point P, what is the angle of rotation (in degrees)?

23 LESSON FIVE - Trigonometric Equations Example Inverse Trigonometric Functions When we solve a trigonometric equation like cosx = -, one possible way to write the solution is: Inverse Trigonometric Functions Enrichment Example Students who plan on taking university calculus should complete this example. In this example, we will explore the inverse functions of sine and cosine to learn why taking an inverse actually yields the solution. a) When we draw the inverse of trigonometric graphs, it is helpful to use a grid that is labeled with both radians and integers. Briefly explain how this is helpful. y x

24 LESSON FIVE - Trigonometric Equations b) Draw the inverse function of each graph. State the domain and range of the original and inverse graphs (after restricting the domain of the original so the inverse is a function). y = sinx y 6 y = cosx y x x c) Is there more than one way to restrict the domain of the original graph so the inverse is a function? If there is, generalize the rule in a sentence. d) Using the inverse graphs from part (b), evaluate each of the following:

25 LESSON SIX - Trigonometric Identities I Example Understanding Trigonometric Identities. a) Why are trigonometric identities considered to be a special type of trigonometric equation? Trigonometric Identities A trigonometric equation that IS an identity: A trigonometric equation that is NOT an identity: b) Which of the following trigonometric equations are also trigonometric identities? i) ii) iii) iv) v)

26 LESSON SIX- Trigonometric Identities I Example The Pythagorean Identities. a) Using the definition of the unit circle, derive the identity sin x + cos x =. Why is sin x + cos x = called a Pythagorean Identity? Pythagorean Identities b) Verify that sin x + cos x = is an identity using i) x = and ii) x =. c) Verify that sin x + cos x = is an identity using a graphing calculator to draw the graph. sin x + cos x = -

27 LESSON SIX - Trigonometric Identities I d) Using the identity sin x + cos x =, derive + cot x = csc x and tan x + = sec x. e) Verify that + cot x = csc x and tan x + = sec x are identities for x =. f) Verify that + cot x = csc x and tan x + = sec x are identities graphically. + cot x = csc x tan x + = sec x

28 LESSON SIX- Trigonometric Identities I a) Example Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities NOTE: You will need to use a graphing calculator to obtain the graphs in this lesson. Make sure the calculator is in RADIAN mode, and use window settings that match the grid provided in each example. Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -

29 LESSON SIX - Trigonometric Identities I a) Example 4 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable)

30 LESSON SIX- Trigonometric Identities I a) Example 5 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -

31 LESSON SIX - Trigonometric Identities I c) Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -

32 LESSON SIX- Trigonometric Identities I a) Example 6 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -

33 LESSON SIX - Trigonometric Identities I c) Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) d) 0 Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -

34 LESSON SIX- Trigonometric Identities I a) Example 7 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable)

35 LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable)

36 LESSON SIX- Trigonometric Identities I a) Example 8 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - b) - Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable)

37 LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable)

38 LESSON SIX- Trigonometric Identities I Example 9 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

39 LESSON SIX - Trigonometric Identities I Example 0 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

40 LESSON SIX- Trigonometric Identities I Example Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

41 LESSON SIX - Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values for. d) Show graphically that Are the graphs exactly the same? y = sinx - y = tanxcosx -

42 LESSON SIX- Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values d) Show graphically that for. Are the graphs exactly the same? y = y =

43 LESSON SIX - Trigonometric Identities I Example 4 Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. d) Show graphically that c) State the the non-permissible values for. Are the graphs exactly the same? y = y =

44 LESSON SIX- Trigonometric Identities I Example 5 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d)

45 LESSON SIX - Trigonometric Identities I Example 6 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d) - -

46 LESSON SIX- Trigonometric Identities I Example 7 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d)

47 LESSON SIX - Trigonometric Identities I Example 8 Use the Pythagorean identities to find the indicated value and draw the corresponding triangle. Pythagorean Identities and Finding an Unknown a) If the value of find the value of cosx within the same domain. b) If the value of, find the value of seca within the same domain. 7 c) If cosθ =, and cotθ < 0, find the exact value of sinθ. 7

48 LESSON SIX- Trigonometric Identities I Example 9 Trigonometric Substitution. Trigonometric Substitution a) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to b. θ a b b) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to a. θ a 4

49 LESSON SEVEN - Trigonometric Identities II Example Evaluate each trigonometric sum or difference. Sum and Difference Identities a) b) c) d) e) f)

50 LESSON SEVEN- Trigonometric Identities II Example Write each expression as a single trigonometric ratio. Sum and Difference Identities a) b) c)

51 LESSON SEVEN - Trigonometric Identities II a) Example Find the exact value of each expression. Sum and Difference Identities b) c) d) Given the exact values of cosine and sine for 5, fill in the blanks for the other angles.

52 LESSON SEVEN- Trigonometric Identities II a) Example 4 Find the exact value of each expression. For simplicity, do not rationalize the denominator. Sum and Difference Identities b) c)

53 LESSON SEVEN - Trigonometric Identities II Example 5 Double-angle identities. Double-Angle Identities a) Prove the double-angle sine identity, sinx = sinxcosx. b) Prove the double-angle cosine identity, cosx = cos x - sin x. c) The double-angle cosine identity, cosx = cos x - sin x, can be expressed as cosx = - sin x or cosx = cos x -. Derive each identity. d) Derive the double-angle tan identity,.

54 LESSON SEVEN- Trigonometric Identities II Example 6 Double-angle identities. Double-Angle Identities 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.

55 LESSON SEVEN - Trigonometric Identities II Example 7 Prove each trigonometric identity. Note: Variable restrictions may be ignored for the proofs in this lesson. Sum and Difference Identities a) b) c) d)

56 LESSON SEVEN- Trigonometric Identities II Example 8 Prove each trigonometric identity. Sum and Difference Identities a) b) c) d)

57 LESSON SEVEN - Trigonometric Identities II Example 9 Prove each trigonometric identity. Double-Angle Identities a) b) c) d)

58 LESSON SEVEN- Trigonometric Identities II Example 0 Prove each trigonometric identity. Double-Angle Identities a) b) c) d)

59 LESSON SEVEN - Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)

60 LESSON SEVEN- Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)

61 LESSON SEVEN - Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)

62 LESSON SEVEN- Trigonometric Identities II Example 4 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)

63 LESSON SEVEN - Trigonometric Identities II Example 5 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)

64 LESSON SEVEN- Trigonometric Identities II Example 6 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d) 0

65 LESSON SEVEN - Trigonometric Identities II Example 7 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)

66 LESSON SEVEN- Trigonometric Identities II Example 8 Trigonometric identities and geometry. A a) Show that B C b) If A = and B = 89, what is the value of C?

67 LESSON SEVEN - Trigonometric Identities II Example 9 Trigonometric identities and geometry. Solve for x. Round your answer to the nearest tenth. x B 5 A

68 LESSON SEVEN- Trigonometric Identities II Example 0 If a cannon shoots a cannonball θ degrees above the horizontal, the horizontal distance traveled by the cannonball before it hits the ground can be found with the function: θ d ( ) θ = v i sinθ cosθ 4.9 The initial velocity of the cannonball is 6 m/s. a) Rewrite the function so it involves a single trigonometric identity. b) Graph the function. Use the graph to describe the trajectory of the cannonball at the following angles: 0, 45, and 90. d θ c) If the cannonball travels a horizontal distance of 00 m, find the angle of the cannon. Solve graphically, and round your answer to the nearest tenth of a degree.

69 LESSON SEVEN - Trigonometric Identities II Example An engineer is planning the construction of a road through a tunnel. In one possible design, the width of the road maximizes the area of a rectangle inscribed within the cross-section of the tunnel. The angle of elevation from the centre line of the road to the upper corner of the rectangle is θ. Sidewalks on either side of the road are included in the design. a) If the area of the rectangle can be represented by the function A(θ) = msinθ, what is the value of m? sidewalk θ 70 m road width sidewalk b) What angle maximizes the area of the rectangular cross-section? c) For the angle that maximizes the area: i) What is the width of the road? ii) What is the height of the tallest vehicle that will pass through the tunnel? iii) What is the width of one of the sidewalks? Express answers as exact values. A θ

70 LESSON SEVEN- Trigonometric Identities II Example The improper placement of speakers for a home theater system may result in a diminished sound quality at the primary viewing area. This phenomenon occurs because sound waves interact with each other in a process called interference. When two sound waves undergo interference, they combine to form a resultant sound wave that has an amplitude equal to the sum of the component sound wave amplitudes. If the amplitude of the resultant wave is larger than the component wave amplitudes, we say the component waves experienced constructive interference. If the amplitude of the resultant wave is smaller than the component wave amplitudes, we say the component waves experienced destructive interference. a) Two sound waves are represented with f(θ) and g(θ). i) Draw the graph of y = f(θ) + g(θ) and determine the resultant wave function. ii) Is this constructive or destructive interference? iii) Will the new sound be louder or quieter than the original sound? 6 g(θ) = 4cosθ f(θ) = cosθ 0-6

71 LESSON SEVEN - Trigonometric Identities II b) A different set of sound waves are represented with m(θ) and n(θ). i) Draw the graph of y = m(θ) + n(θ) and determine the resultant wave function. ii) Is this constructive or destructive interference? iii) Will the new sound be louder or quieter than the original sound? 6 m(θ) = cosθ 0 n(θ) = cos(θ - ) -6 c) Two sound waves experience total destructive interference if the sum of their wave functions is zero. Given p(θ) = sin(θ - /4) and q(θ) = sin(θ - 7/4), show that these waves experience total destructive interference.

72 LESSON SEVEN- Trigonometric Identities II Example Even & Odd Identities Even & Odd Identities a) Explain what is meant by the terms even function and odd function. b) Explain how the even & odd identities work. (Reference the unit circle or trigonometric graphs in your answer.) c) Prove the three even & odd identities algebraically.

73 LESSON SEVEN - Trigonometric Identities II Example 4 Proving the sum and difference identities. a) Explain how to construct the diagram shown. Enrichment Example Students who plan on taking university calculus should complete this example. D C F β α A E G B b) Explain the next steps in the construction. D α C H F β α A E G B

74 LESSON SEVEN- Trigonometric Identities II c) State the side lengths of all the triangles. D D D α H F F α+β β F A E A A α G d) Prove the sum and difference identity for sine.

75 Answer Key Trigonometry Lesson Five: Trigonometric Equations Note: n ε I for all general solutions. Example : a), b), c) d) Example : a) b) c) d) no solution e) f) Example : a) b) c) Example 4: a) b) Intersection point(s) of original equation θ-intercepts Intersection point(s) of original equation θ-intercepts

76 Answer Key Example 5: a) b) c) d) Example 6: a) and 4.54 b) and 4.54 c) and 4.54 d) and 4.54 The unit circle is not useful for this question Example 7: - - a) b) c) Example 8: a) No Solution b) c) d) e) f)

77 Answer Key Example 9: a) b) c) Example 0: a) No Solution b) Intersection point(s) of original equation θ-intercepts Intersection point(s) of original equation θ-intercepts Example : a) b) c) d) Example : a) b) c) d) The unit circle is not useful for this question Example : - a) b) c) d)

78 Answer Key Example 4: a) b) c) d) Example 5: a) b) c) d) Example 6: a) b) c) Example 7: Example 8: a) b) a) b) Example 9: a) Example 0: a) Example : See Video b) Approximately days. Visible % b) See graph. c) rad (or 6.6 ) d θ t -8

79 Answer Key Trigonometry Lesson Six: Trigonometric Identities I Note: n ε I for all general solutions. Example : a) Identity Equation b) i) ii) iii) iv) v) Not an Identity Not an Identity Identity Identity Not an Identity Example : a) Use basic trigonometry (SOHCAHTOA) to show that x = cosθ and y = sinθ. θ x y b) Verify that the L.S. = R.S. for each angle. c) The graphs of y = sin x + cos x and y = are the same. - d) Divide both sides of sin x + cos x = by sin x to get + cot x = csc x. Divide both sides of sin x + cos x = by cos x to get tan x + = sec x. Example : a) b) e) Verify that the L.S. = R.S. for each angle. f) The graphs of y = + cot x and y = csc x are the same Example 5: a), b) The graphs of y = tan x + and y = sec x are the same Example 4: a) b) - - c) d)

80 Answer Key Example 6: a) b) - - c) d) 0 - Example 7: a) b) c) d) Example 8: a) b) c) d)

81 Answer Key Example 9: See Video Example 0: See Video Example : See Video Example : a) See Video b) c) d) Example : a) See Video b) c) d) Example 4: a) See Video b) c) d) The graphs are NOT identical. The R.S. has holes. The graphs are identical. The graphs are identical Example 5: a), b), c), d), Example 6: a), b), - 0 Note: All terms from the original equation were collected on the left side before graphing c), d), - -

82 Answer Key Example 7: a), b), 0 Note: All terms from the original equation were collected on the left side before graphing Note: All terms from the original equation were collected on the left side before graphing c), d), Example 8: a) b) c) Example 9: See Video

83 Answer Key Trigonometry Lesson Seven: Trigonometric Identities II Note: n ε I for all general solutions. Example : a) b) c) d) e) f) Example : Example : a) b) c) a) b) c) d) See Video Example 4: Example 5: See Video a) b) c) Example 6: a) i. ii. 0 iii. undefined b) (answers may vary) i. c) (answers may vary) i. ii. iii. iv. ii. iii. iv. Example 0: a) b) d θ -. c) θ = 4.6 and θ = 65.4 Example : At 0, the cannonball hits the ground as soon as it leaves the cannon, so the horizontal distance is 0 m. At 45, the cannonball hits the ground at the maximum horizontal distance,. m. At 90, the cannonball goes straight up and down, landing on the cannon at a horizontal distance of 0 m Examples 7 - : Proofs. See Video. a) Example 4: a) b) Example 5: a) b) b) A 4900 The maximum area occurs when θ = 45. At this angle, the rectangle is the top half of a square. c) c) θ d) d) c) i. ii. iii. Example : Example 6: a) Example 7: a) a) i. y = f(θ) + g(θ) b) i. 6 6 y = f(θ) + g(θ) b) b) 0 0 c) c) d) Example 8: 57 Example 9: 9.9 d) -6 ii. The waves experience constructive interference. iii. The new sound will be louder than either original sound. c) All of the terms subtract out leaving y = 0, A flat line indicating no wave activity. -6 ii. The waves experience destructive interference. iii. The new sound will be quieter than either original sound. Example : See Video. Example 4: See Video.

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85 Mathematics 0- Trigonometry I The Unit Circle Formula Sheet Trigonometry II Note: The unit circle is NOT included on the official formula sheet. Transformations & Operations Exponential and Logarithmic Functions Permutations & Combinations Polynomial, Radical & Rational Functions