PreCalculus 3 rd and 4 th Nine-Weeks Scope and Sequence

Transcription

1 PreCalculus 3 rd and 4 th Nine-Weeks Scope and Sequence Topic 4: (45 50 days) A) Uses radian and degree angle measure to solve problems and perform conversions as needed. B) Uses the unit circle to explain the circular properties and periodic nature of trigonometric functions and to find the trigonometric ratios of any angle. C) Describes and compares the characteristics of the trigonometric functions (with and without the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant. D) Determines solutions to trigonometric equations. E) Describes how a change in the value of any constant in a general-form trigonometric equation such as y = a sin (b-x) + c affects the graph of the equation. F) Represents the inverse of a trigonometric function symbolically and graphically G) Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data, and uses that model to identify patterns and make predictions. H) Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction, and double-angle. I) Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. J) Models and solves problems using trigonometry. Topic 5: Noncartesian Representations (35 40 days) A) Uses vectors to model and solve application problems. B) Uses parametric equations to model and solve application problems. C) Uses polar coordinates. D) Expresses complex numbers in trigonometric form and computes sums, differences, products, quotients, powers, and roots of complex numbers in trigonometric form.

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3 COLUMBUS PUBLIC SCHOOLS MATHEMATICS CURRICULUM GUIDE SUBJECT STATE STANDARD 3 and 4 TIME RANGE GRADING PreCalculus Patterns, Functions, and Algebra, days PERIOD Geometry and Spatial Sense 3-4 MATHEMATICAL TOPIC 4 CPS LEARNING GOALS A) Uses radian and degree angle measure to solve problems and perform conversions as needed. B) Uses the unit circle to explain the circular properties and periodic nature of trigonometric functions and to find the trigonometric ratios of any angle. C) Describes and compares the characteristics of the trigonometric functions (with and without the use of technology) for sine, cosine, tangent, cotangent, cosecant, and secant. D) Determines solutions to trigonometric equations. E) Describes how a change in the value of any constant in a general-form trigonometric equation such as y = a sin (b-x) + c affects the graph of the equation. F) Represents the inverse of a trigonometric function symbolically and graphically G) Creates a scatterplot of bivariate data, identifies a trigonometric function to model the data, and uses that model to identify patterns and make predictions. H) Derives and applies the basic trigonometric identities; i.e. angle addition, angle subtraction, and double-angle. I) Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. J) Models and solves problems using trigonometry. COURSE LEVEL INDICATORS Course Level (i.e., How does a student demonstrate mastery?): Converts between degrees and radians and can explain the appropriate use of each. Math M:11-B:0 Sets up and solves angular velocity and arc-length problems by using radian measure. Math G:1-D:0 Describes and compares the characteristics of the trigonometric functions; e.g., general shape, number of roots, domain and range, even or odd asymptotic and global behavior for sine, cosine, tangent, cotangent, cosecant, and secant both algebraically and graphically. Math A:1-A:03 Relates a given sine or cosine graph to its equivalent other in terms of phase shift. Math A:1-A:03 Determines all zeros for a trigonometric function algebraically and also gives all zeros within a given range (such as between 0 and π radians). Math A:1-A:03 Identifies the amplitude, frequency/period, phase shift, vertical shift, etc. of a given trigonometric function and uses these to sketch a graph of the function. Math A:1-A:03 Identifies the extrema of trigonometric functions with and without technology. Math A:1-A:03 Page 1 of 163 Columbus Public Schools 1/5/06

4 Uses the unit circle to explain the circular properties and periodic nature of trigonometric functions and to find the trigonometric ratios of any angle. Math G:11-A:04 Uses the unit circle to explain the periodic nature of the sine and cosine functions, the nature of reference angles, and the range of possible values for sine and cosine. Math A:1-A:03 Represents the inverse of a trigonometric function symbolically and graphically. Math A:1-A:04 Plots bivariate data and determines the trigonometric function that best fits the data both analytically and by regression. Math D:11-A:04 Determines phase shift, vertical shift, amplitude and frequency to be able to create the trigonometric function equation best fitting the data. Math A:1-A:03 Collects real world motion data and models it using trigonometric equations. Math D:11-C:04 Verifies identities analytically by applying fundamental trigonometric identities to re-write and combine expressions. Math G:1-A:0 Verifies trigonometric identities graphically. Math G:1-A:0 Uses the double-angle, half-angle, and angle-addition formulas to determine specified trigonometric values. Math G:11-A:04 Uses trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines. Math G:11-A:04 Uses the Law of Sines and the Law of Cosines to determine the missing angles or sides of triangles. Math G:11-A:04 Uses Heron s formula to find the area of a triangle when the sides are known but base and/or height are not given. Math G:11-A:04 Determines general solutions to trigonometric equations and specific solutions within a given interval. Math G:11-A:04 Previous Level: Defines the basic trigonometric ratios in right triangles: sine, cosine, and tangent. Math G:09-I:01 Uses right triangle trigonometric relationships to determine lengths and angle measures. Math G:09-I:0 Evaluates expressions containing square roots. Math N:09-I:04 Sketches a basic sine and cosine graph by hand. Math A:10-D:0 Sketches the graph of a function by means of technology. Math A:08-D:09 Next Level: Determines the average rate of change for specific trigonometric functions. Math A:1-A:10 Page of 163 Columbus Public Schools 1/5/06

5 The description from the state for the Measurement Standard says: Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies. The grade-band benchmark from the state, for this topic in the grade band 11 1 is: B. Apply various measurement scales to describe phenomena and solve problems. The description from the state, for the Geometry and Spatial Sense Standard says: Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two-, and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems. The grade-band benchmarks from the state, for this topic in the grade band 11 1 are: A. Use trigonometric relationships to verify and determine solutions in problem situations. D. Use coordinate geometry to represent and examine the properties of geometric figures.* The description from the state, for the Patterns, Functions, and Algebra Standard says: Students use patterns, relations, and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations. The grade-band benchmark from the state, for this topic in the grade band 11 1 is: A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes and local and global behavior. The description from the state for the Data Analysis Standard says: Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. The grade-band benchmarks from the state, for this topic in the grade band 11 1 are: A. Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators. C. Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions. The description from the state, for the Mathematical Processes Standard says: Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The grade-band benchmark from the state, for this topic in the grade band 11 1 is: J. Apply mathematical modeling to workplace and consumer situations including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation. *This is an extension of the benchmarks in grades 8-10 for more complex figures. Page 3 of 163 Columbus Public Schools 1/5/06

6 PRACTICE ASSESSMENT ITEMS Trigonometry - A A wheel rotating at 50 revolutions per minute rotates at A. 50π radians/minute B. 100π radians/minute C. 150π radians/minute D. 00π radians/minute Which of the following gives the measures of two angles, one positive and one negative, that are coterminal with a 73 degree angle? A. 433 o, -87 o B. 433 o, -107 o C. 163 o, -17 o D. 53 o, -107 o Which of the following expressions can be used to calculate the length of an arc that subtends a central angle of measure 65º on a circle of diameter 30 meters? π A meters 180º 180 B meters π C meters D. 65 meters Page 4 of 163 Columbus Public Schools 1/5/06

7 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - A Low Complexity A wheel rotating at 50 revolutions per minute rotates at A. 50π radians/minute B. 100π radians/minute C. 150π radians/minute D. 00π radians/minute Answer: B Which of the following gives the measures of two angles, one positive and one negative, that are coterminal with a 73 degree angle? A. 433 o, -87 o B. 433 o, -107 o C. 163 o, -17 o D. 53 o, -107 o Moderate Complexity Answer: A Which of the following expressions can be used to calculate the length of an arc that subtends a central angle of measure 65º on a circle of diameter 30 meters? π A meters 180º 180 B meters π C meters D. 65 meters Answer: A Page 5 of 163 Columbus Public Schools 1/5/06

8 PRACTICE ASSESSMENT ITEMS Trigonometry - A A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes 70 seconds to complete one revolution. Which of the following correctly determines the angular speed ω and the linear speed v of the Ferris wheel? A. ω = θ t = πrad 70sec =.045rad / sec, v = s t B. ω = θ t = πrad 70sec =.09rad /sec, v = s t C. ω = θ t 30 ft = = 0.43 ft / sec, 70sec v = s t D. ω = θ t = πrad 70sec =.09rad /sec, v = s t = π 30 ft 70sec =.7 ft /sec 30 ft = = 0.43 ft /sec 70sec 30 ft = = 0.43 ft /sec 70sec = π 30 ft 70sec =.7 ft /sec A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135º a. Determine the area of the lawn that receives water. b. If we wanted the area of coverage to be exactly 1500 square feet, determine we should change the radius to, keeping the angle the same. It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 5.8 inches shorter than the outside arc, determine the width of the track. Page 6 of 163 Columbus Public Schools 1/5/06

9 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - A High Complexity A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure that it takes 70 seconds to complete one revolution. Which of the following correctly determines the angular speed ω and the linear speed v of the Ferris wheel? A. ω = θ t = πrad 70sec =.045rad / sec, v = s t B. ω = θ t = πrad 70sec =.09rad /sec, v = s t C. ω = θ t 30 ft = = 0.43 ft / sec, 70sec v Short Answer/Extended Response = s t D. ω = θ t = πrad 70sec =.09rad /sec, v = s t = π 30 ft 70sec =.7 ft /sec 30 ft = = 0.43 ft /sec 70sec 30 ft = = 0.43 ft /sec 70sec = π 30 ft 70sec Answer: D =.7 ft /sec A water sprinkler sprays water over a distance of 30 feet while rotating through an angle of 135º a. Determine the area of the lawn that receives water. b. If we wanted the area of coverage to be exactly 1500 square feet, determine we should change the radius to, keeping the angle the same. Solution π radians 3π a. 135 = radians A = r θ = 30 π 4 = ft b. A = r θ 1500 = r π r = r = ft 4 A 4 point response: shows work and gets correct answers to both parts. A 3 point response gets one solution correct, and the other has a single error in substitution or calculation. A point response gets the first part right, but does not get the proper set-up for the second part. A 1 point response has a major conceptual error on the first part, and does not show any understanding of the second part. A 0 point response shows no understanding of the problem. Page 7 of 163 Columbus Public Schools 1/5/06

10 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - A Short Answer/Extended Response It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 5.8 inches shorter than the outside arc, determine the width of the track. Solution: x+5.8 x r w π Each piece of track has an angle measure of 10 (or π 5 ) radians. π The arc length of the inside of one piece we re taking as x, so x = 5 r π The arc length of the outside of one piece will then be: x = 5 (r + w) To solve both of these equations simultaneously, we multiply both equations by 5 and get: 5x = πr and 5x + 9 = π(r + w) 9 Solving these, we get w = = 9.3 inches π A 4-point response uses the arc-length formulas to set-up and solve simultaneous equations and gets the correct width. A 3-point response has appropriate set-up for the two arc-length formulas but doesn t reach a proper solution for the width. A -point response specifies the arc-length formula but doesn t properly represent the 5.8- inch difference and does not determine the width. A 1-point response illustrates all necessary components of the problem but goes no further. A 0-point response shows no understanding of the problem. Page 8 of 163 Columbus Public Schools 1/5/06

11 PRACTICE ASSESSMENT ITEMS Trigonometry - B Which of the following gives the range of the function f(x) = (sin x) +(cos x)? A. {0} B. {1} C. [0, 1] D. [ 1, 1] E. [0, ] A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value? A. -1 B -10 C. -9 D. 0 E. 3 Which of the following functions have identical graphs? (i) y = sin( x + π 6) (ii) y = cos( x π 6) (iii) y = cos( x π 3) A) i and iii B) ii and iii C) i and ii D) i, ii and iii Page 9 of 163 Columbus Public Schools 1/5/06

12 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - B Low Complexity Which of the following gives the range of the function f(x) = (sin x) +(cos x)? A. {0} B. {1} C. [0, 1] D. [ 1, 1] E. [0, ] Answer: B A sinusoid with amplitude 6 has a maximum value of 3. What is its minimum value? A. -1 B. -10 C. -9 D. 0 E. 3 Moderate Complexity Answer: C Which of the following functions have identical graphs? (i) y = sin( x + π 6) (ii) y = cos( x π 6) (iii) y = cos( x π 3) A) i and iii B) ii and iii C) i and ii D) i, ii and iii Answer: A Page 10 of 163 Columbus Public Schools 1/5/06

13 PRACTICE ASSESSMENT ITEMS Trigonometry - B Which of the following trigonometric functions are odd? i. y = sin(x) ii. y = cos(x) iii. y = tan(x) A. i and ii only B. ii and iii only C. i and iii only D. i, ii, and iii Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be approximated by T = 40sin π 365 t where t is in days with t = 0 corresponding to January 1. Predict the date when the coldest day of the year will occur and give the temperature for that day. Show your solution. Page 11 of 163 Columbus Public Schools 1/5/06

14 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - B High Complexity Which of the following trigonometric functions are odd? i. y = sin(x) ii. y = cos(x) iii. y = tan(x) A. i and ii only B. ii and iii only C. i and iii only D. i, ii, and iii Short Answer/Extended Response Answer: C Based on years of weather data in a certain city, the expected low temperature T (in ºF) can be approximated by T = 40sin π 365 t where t is in days with t = 0 corresponding to January 1. Predict the date when the coldest day of the year will occur and give the temperature for that day. Show your solution. Solution: By using the minimum function on the graphing calculator, we get minimum temperatures of -5 o on day 4 (January 4). A -point response discusses proper use of the graph to get the correct answer. A 1-point response uses a graph to get an incorrect answer. A 0-point response shows no mathematical understanding of the task. Page 1 of 163 Columbus Public Schools 1/5/06

15 PRACTICE ASSESSMENT ITEMS Trigonometry - C π is the reference angle for which one of the following non-acute angles? 6 A) π 3 B) π 3 C) 5π 6 D) 4π 3 Which of the following must have the same value as cos(68º)? A. sin(º) B. cos(º) C. tan(º) D. none of the above Page 13 of 163 Columbus Public Schools 1/5/06

16 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - C Low Complexity π is the reference angle for which one of the following non-acute angles? 6 A) π 3 B) π 3 C) 5π 6 D) 4π 3 Answer: C Moderate Complexity Which of the following must have the same value as cos(68º)? A. sin(º) B. cos(º) C. tan(º) D. none of the above Answer: A Page 14 of 163 Columbus Public Schools 1/5/06

17 PRACTICE ASSESSMENT ITEMS Trigonometry - C If cosθ = 1 13 A B. 5 1 and tanθ < 0, then sinθ =? C D What would be the coordinates of the point P where the terminal ray of an angle θ intersects the unit circle? A. (sinθ,cosθ) B. (cosθ,sinθ) C. (r, r ) P D. (r,θ) r=1 θ Page 15 of 163 Columbus Public Schools 1/5/06

18 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - C High Complexity If cosθ = 1 13 A B. 5 1 and tanθ < 0, then sinθ =? C D Answer: D High Complexity What would be the coordinates of the point P where the terminal ray of an angle θ intersects the unit circle? A. (sinθ,cosθ) B. (cosθ,sinθ) C. (r, r ) P D. (r,θ) r=1 θ Answer: B Page 16 of 163 Columbus Public Schools 1/5/06

19 PRACTICE ASSESSMENT ITEMS Trigonometry - C Evalute cos 7π 6 without using a calculator by using ratios and the relevant reference triangle. Show all work. Page 17 of 163 Columbus Public Schools 1/5/06

20 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - C Short Answer/Extended Response Evalute cos 7π 6 without using a calculator by using ratios and the relevant reference triangle. Show all work. Solution: 7π is in the third quadrant where cosine is negative. The reference angle for the third 6 quadrant is 7π 6 π, or π 6. Thus we need cos π 3 which equals 6 A -point response properly explains where we get the sign of the answer and how we determine the reference angle, and gets the correct answer. A 1-point response makes at most one error (sign, reference angle, or result of trig function). A 0-point response shows no mathematical understanding of the task. Page 18 of 163 Columbus Public Schools 1/5/06

21 PRACTICE ASSESSMENT ITEMS Trigonometry - D Which one of the following trigonometric functions has its graph symmetric about the line y = - 6? A) y = 6sin(3x) B) y = 8sin(x 6) + 1 C) y = 4cos(x) 6 x D) y = sin 6 What is the phase shift of y = 5sin(x 3π )? A) 3π to the left B) 3π to the right C) 3π to the left D) 3π to the right Page 19 of 163 Columbus Public Schools 1/5/06

22 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - D Low Complexity Which one of the following trigonometric functions has its graph symmetric about the line y = - 6? A) y = 6sin(3x) B) y = 8sin(x 6) + 1 C) y = 4cos(x) 6 x D) y = sin 6 Answer: C Moderate Complexity What is the phase shift of y = 5sin(x 3π )? A) 3π to the left B) 3π to the right C) 3π to the left D) 3π to the right Answer: C (Trick: This is really y = 5sin(x - 3 π ) Page 0 of 163 Columbus Public Schools 1/5/06

23 PRACTICE ASSESSMENT ITEMS Trigonometry - D Which equation matches the graph below? A) y = cos4( x π) B) y = cos( 4x π) C) y = cos4( x + π) D) y = cos( 4x + π) State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift and vertical shift: y = 6cos x Page 1 of 163 Columbus Public Schools 1/5/06

24 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - D High Complexity Which equation matches the graph below? A) y = cos4( x π) B) y = cos( 4x π) C) y = cos4( x + π) D) y = cos( 4x + π) Answer: D Short Answer/Extended Response State the amplitude and period of the sinusoid and (relative to the parent function) the phase shift and vertical shift: y = 6cos x Solution: Cosine graph, shifted left 4, multiply period by (horizontal stretch, yielding a period of π or 4π, multiply amplitude by 6, and shift down by 3. 1 A point response correctly determines all information asked for. A 1 point response gets at least pieces of the question correct but shows a lack of understanding for the other two. A 0 point response shows no mathematical understanding of the problem. Page of 163 Columbus Public Schools 1/5/06

25 PRACTICE ASSESSMENT ITEMS Trigonometry - E The exact value of arctan 3 is: A. 0 B. π 6 C. π 4 D. π 3 The range of the function f (x) = arcsin x is: A. (, ) B. ( 1,1) C. [ 1,1] D. [0,π] E. [ π /, π /] Page 3 of 163 Columbus Public Schools 1/5/06

26 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - E Low Complexity The exact value of arctan 3 is: A. 0 B. π 6 C. π 4 D. π 3 Moderate Complexity Answer: D The range of the function f (x) = arcsin x is: A. (, ) B. ( 1,1) C. [ 1,1] D. [0,π] E. [ π /, π /] Answer: E Page 4 of 163 Columbus Public Schools 1/5/06

27 PRACTICE ASSESSMENT ITEMS Trigonometry - E sec(tan 1 (x)) = A. x B. csc x C. 1+ x D. 1 x E. sin x cos x You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice daily with approximate regularity. Remembering that the trigonometric functions model repetitive behavior, you place a meter stick in the water to measure water height every hour between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high tide the height is 80 centimeters. Determine a reasonable defining equation for this function and explain how you determined your answer. Page 5 of 163 Columbus Public Schools 1/5/06

28 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - E High Complexity sec(tan 1 (x)) = A. x B. csc x C. 1+ x D. 1 x E. sin x cos x Short Answer/Extended Response Answer: C You rent a cottage on the ocean for a week one summer and notice that the tide comes in twice daily with approximate regularity. Remembering that the trigonometric functions model repetitive behavior, you place a meter stick in the water to measure water height every hour between 6:00 AM to midnight. At low tide the height of the water is 0 centimeters and at high tide the height is 80 centimeters. Determine a reasonable defining equation for this function and explain how you determined your answer. Student Answers Will Vary. A sample solution is: Amplitude: a = 1 (80 0) = 40. Period: π π = 1, therefore b =. b 6 c π Displacement: 3 =, therefore c =. Vertical Shift = d = 40. b π π y = 40 sin x A point response correctly determines all information asked for. A 1 point response gets at least pieces of the question correct but shows a lack of understanding for the other two. A 0 point response shows no mathematical understanding of the problem. Page 6 of 163 Columbus Public Schools 1/5/06

29 PRACTICE ASSESSMENT ITEMS Trigonometry - F Which statement is always true for the inverse of every trigonometric function? A. The inverse of every trigonometric function has a range of [-1, 1]. B. The inverse of every trigonometric function passes the vertical line test. C. The inverse of every trigonometric function has a domain of [0, π]. D. The inverse of every trigonometric function is not a function. Which graph represent the equation y = cos -1 x? A. 4 B. 4 C. 4 D Page 7 of 163 Columbus Public Schools 1/5/06

30 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - F Low Complexity Which statement is always true for the inverse of every trigonometric function? A. The inverse of every trigonometric function has a range of [-1, 1]. B. The inverse of every trigonometric function passes the vertical line test. C. The inverse of every trigonometric function has a domain of [0, π]. D. The inverse of every trigonometric function is not a function. Moderate Complexity Answer: D Which graph represent the equation y = cos -1 x? A. B. C. D Answer: C Page 8 of 163 Columbus Public Schools 1/5/06

31 PRACTICE ASSESSMENT ITEMS Trigonometry - F Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small town are shown in the table. Month Temp ( o F) Model this data using f(x) = a cos(b(x-c)) + d A. f (x) = 16cos( π (x 3) 6 )+ 53 B. f (x) = 1cos( π (x 3) 6 )+ 53 C. f (x) = 16cos( π x 6 )+ 53 D. f (x) = 16cos( π (x + 3) 6 )+ 53 Tides go up and down in a 1.-hour period. The average depth of a certain river is 14m and ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation that gives the approximate variation y, if x is the number of hours after midnight with high tide occurring at 8 am. Justify each part of your answer using relevant terminology. Page 9 of 163 Columbus Public Schools 1/5/06

32 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - F High Complexity Suppose that the average monthly low temperatures (rounded to the nearest degree) for a small town are shown in the table. Month Temp ( o F) Model this data using f(x) = a cos(b(x-c)) + d A. f (x) = 16cos( π (x 3) 6 )+ 53 B. f (x) = 1cos( π (x 3) 6 )+ 53 C. f (x) = 16cos( π x 6 )+ 53 D. f (x) = 16cos( π (x + 3) 6 )+ 53 Answer: A Page 30 of 163 Columbus Public Schools 1/5/06

33 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - F Short Answer/Extended Response Tides go up and down in a 1.-hour period. The average depth of a certain river is 14m and ranges from 9 to 19m. The variation can be approximated by a sine curve. Write an equation that gives the approximate variation y, if x is the number of hours after midnight with high tide occurring at 8 am. Justify each part of your answer using relevant terminology. Answer: The period is 1., so the coefficient for x would be π π, or The average depth of 14 gives our vertical shift of 14. The starting point being hour 8 means that 8 must be subtracted from x in the equation (phase shift of 8 hours to the left from the parent graph) Half the difference between our maximum, 19m, and our minimum, 9 m, gives us our amplitude of 5. π y = 5sin ( x 8) A 4 point response includes proper explanations for the period, vertical shift, period, phase shift, and amplitude, and puts them together into the equation given above. A 3 point response has one mistake in one of the four properties described above but shows a proper understanding of the other three. A point response has flaws in two of the four modifications to the parent function, but properly represents the other two in the equation. A 1 point response shows understanding of one aspect of the data in terms of modifying the parent function. A 0 point response shows no mathematical understanding of the problem. Page 31 of 163 Columbus Public Schools 1/5/06

34 PRACTICE ASSESSMENT ITEMS Trigonometry - G 4 3 If sin( x ) = and cos( x ) =, then sin(x) and cos(x) would be: A. sin( x) =, cos( x) = B. sin( x) =, cos( x) = C. sin( x) =, cos( x) = D. sin( x) =, cos( x) = 5 5 Which one of the following is equivalent to tanθ + secθ? A. sinθ + cosθ B. tanθ + cscθ C. sin θ + 1 cosθ D. cosθ 1+ sinθ E. cosθ cotθ Page 3 of 163 Columbus Public Schools 1/5/06

35 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - G Low Complexity 4 3 If sin( x ) = and cos( x ) =, then sin(x) and cos(x) would be: A. sin( x) =, cos( x) = B. sin( x) =, cos( x) = C. sin( x) =, cos( x) = D. sin( x) =, cos( x) = 5 5 Answer: A Moderate Complexity Which one of the following is equivalent to tanθ + secθ? A. sinθ + cosθ B. tanθ + cscθ C. sin θ + 1 cosθ D. cosθ 1+ sinθ E. cosθ cotθ Answer: C Page 33 of 163 Columbus Public Schools 1/5/06

36 PRACTICE ASSESSMENT ITEMS Trigonometry - G Which of the following gives the expression A. csc x B. -cot x C. -tan x D. sec x cos x cos x 1+ cos x 1 cos x in completely simplified form? Prove the identity: x x+ x x = x+ x (cos )(tan sin cot ) sin cos Page 34 of 163 Columbus Public Schools 1/5/06

37 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - G High Complexity Which of the following gives the expression A. csc x B. -cot x C. -tan x D. sec x cos x cos x 1+ cos x 1 cos x in completely simplified form? Answer: B Short Answer/Extended Response Prove the identity: x x+ x x = x+ x (cos )(tan sin cot ) sin cos Answers Will Vary. A sample solution is: sin x cos x (cos x) + sin x sin x cos x cos x sin x = + sin x (cos x) + cos x = sin x+ cos x cos x sin + cos = sin + cos x x x x Page 35 of 163 Columbus Public Schools 1/5/06

38 PRACTICE ASSESSMENT ITEMS Trigonometry - H Two boats starting at the same place and time, speed away along courses that form a 150º angle. If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine how far apart the boats are after 0 minutes? A mi B mi C mi D mi E mi Given a triangle with the following information provided: a = 38, b = 19, C = 1º, which of the following would we use first in order to solve the triangle: A. The Pythagorean Theorem B. The Law of Sines C. The Law of Cosines D. The Quadratic Formula Page 36 of 163 Columbus Public Schools 1/5/06

39 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - H Low Complexity Two boats starting at the same place and time, speed away along courses that form a 150º angle. If one boat travels at 54 miles per hour and the other boat travels at 30 miles per hour, determine how far apart the boats are after 0 minutes? A mi B mi C mi D mi E mi Moderate Complexity Answer: D Given a triangle with the following information provided: a = 38, b = 19, C = 1º, which of the following would we use first in order to solve the triangle: A. The Pythagorean Theorem B. The Law of Sines C. The Law of Cosines D. The Quadratic Formula Answer: C Page 37 of 163 Columbus Public Schools 1/5/06

40 PRACTICE ASSESSMENT ITEMS Trigonometry - H Given the following information about a triangle: a = 9, b = 5, c = 10, and the measure of angle B = 93 o Which one of the following is true? A. Side-lengths a, b, and c don t satisfy the Triangle Inequality. B. We can use the Law of Sines to determine angles A or C. C. We cannot have a triangle since the longest side is not opposite the largest angle. D. The three side-lengths form a Pythagorean triple. Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C. Page 38 of 163 Columbus Public Schools 1/5/06

41 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - H High Complexity Given the following information about a triangle: a = 9, b = 5, c = 10, and the measure of angle B = 93 o Which one of the following is true? A. Side-lengths a, b, and c don t satisfy the Triangle Inequality. B. We can use the Law of Sines to determine angles A or C. C. We cannot have a triangle since the longest side is not opposite the largest angle. D. The three side-lengths form a Pythagorean triple. Short Answer/Extended Response Answer: C Given a triangle where a = 55, c = 80, and A = 35º, find two possible values for angle C. sin 35º sinc = sin 35 C = sin 55 =56.54º,13.46º A 4-point answer clearly shows correct use of the Law of Sines to get correct values. A 3-point answer shows a correct solution for the 1 st angle but makes a minor computation error in determining the second angle. A -point answer shows correct set-up and work to find only the first angle. A 1-point answer sets up the proper Law of Sines proportion but fails to get a single correct answer. A 0-point answer shows no understanding of the problem. Page 39 of 163 Columbus Public Schools 1/5/06

42 PRACTICE ASSESSMENT ITEMS Trigonometry - I A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of inclination of the ramp to the nearest tenth of a degree? A o B. 4.0 o C o D o To approximate the height of a radio tower, Mark counts off 7 feet from the base of the tower and then measures the angle of elevation from the ground to the top of the tower from that point to be 40 o. Approximately how tall is the tower? A ft B. 55. ft C ft D. 7 ft Page 40 of 163 Columbus Public Schools 1/5/06

43 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - I Low Complexity A ramp leading to a freeway overpass is 540 ft long and rises 38 ft. What is the average angle of inclination of the ramp to the nearest tenth of a degree? A o B. 4.0 o C o D o Moderate Complexity Answer: B To approximate the height of a radio tower, Mark counts off 7 feet from the base of the tower and then measures the angle of elevation from the ground to the top of the tower from that point to be 40 o. Approximately how tall is the tower? A ft B. 55. ft C ft D. 7 ft Answer: C Page 41 of 163 Columbus Public Schools 1/5/06

44 PRACTICE ASSESSMENT ITEMS Trigonometry - I A boat leaves harbor and travels at 30 knots on a bearing of 83 o. After four hours, it changes course to a bearing of 138 o and continues at 30 knots for three hours. After the entire seven-hour trip, how far is the boat from its starting point? A nautical miles B nautical miles C nautical miles D nautical miles From the top of a 5 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work. Page 4 of 163 Columbus Public Schools 1/5/06

45 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - I High Complexity A boat leaves harbor and travels at 30 knots on a bearing of 83 o. After four hours, it changes course to a bearing of 138 o and continues at 30 knots for three hours. After the entire seven-hour trip, how far is the boat from its starting point? A nautical miles B nautical miles C nautical miles D nautical miles Answer: C Page 43 of 163 Columbus Public Schools 1/5/06

46 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - I Short Answer/Extended Response From the top of a 5 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work. From the top of a 5 ft. building, a man observes a car moving towards the building. If the angle of depression from the man to the car changes from 18º to 39º during the period of observation, determine how far the car travels. Include a diagram to back up your work. 5 ft 18 o 1 o 39 o time time 1 5 ft Time1 7 o 5 ft Time 51 o tan 7 = x 5 x = 5 tan 7 =69.41ft note: you could also solve tan18 = 5 x x y tan 51 = y 5 y = 5 tan 51 =77.85 ft note: you could also solve tan 39 = 5 y Change in Distance = ft ft = ft. A 4-point answer shows both tangent problems set-up and solved properly and gets the correct difference. A 3-point answer gets both the setups and substitutions correct, but makes one mistake in the calculations somewhere along the way. A -point answer sets up the diagram(s) properly and shows the need to find a difference but is unable to use an appropriate trig ratio to determine these distances. A 1-point answer attempts to set up the diagram(s) properly so as to get the necessary angle measures, but is missing crucial pieces. A 0-point answer shows no mathematical understanding of the problem. Page 44 of 163 Columbus Public Schools 1/5/06

47 PRACTICE ASSESSMENT ITEMS Trigonometry - J What is the factored form for the expression 1 cos 3 x? A. (1 cos x) (1 + cos x + cos x) B. (1 cos x) 3 C. (1 cos x) (sin x + cos x) D. (1 cos x) (1 cos x + cos x) Find all solutions to the equation sin x sin x = in the interval[0, π ]. A. B. C. D. π 3π x =, π 3π x =, π, π x = x = 3π Page 45 of 163 Columbus Public Schools 1/5/06

48 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - J Low Complexity What is the factored form for the expression 1 cos 3 x? A. (1 cos x) (1 + cos x + cos x) B. (1 cos x) 3 C. (1 cos x) (sin x + cos x) D. (1 cos x) (1 cos x + cos x) Moderate Complexity Answer: A Find all solutions to the equation sin x sin x = in the interval[0, π ]. A. B. C. D. π 3π x =, π 3π x =, π, π x = x = 3π Answer: D Page 46 of 163 Columbus Public Schools 1/5/06

49 PRACTICE ASSESSMENT ITEMS Trigonometry - J 1 Which values of t on [ π, π] satisfy cost <? A. B. C. D. 4 π 4, π π, π π 4, π π, π π 4π, 3 3 π π, 3 3 x 1+ cosx Use a graphing calculator to find all solutions to the equation tan = in the 1 cos x interval[0, π ]. Include a sketch of your graphs as well as an explanation of how you reached your answer. Page 47 of 163 Columbus Public Schools 1/5/06

50 PRACTICE ASSESSMENT ITEMS Answers/Rubrics Trigonometry - J High Complexity 1 Which values of t on [ π, π] satisfy cost <? A. B. C. D. 4 π 4, π π, π π 4, π π, π π 4π, 3 3 π π, 3 3 Answer: A Short Answer/Extended Response x 1+ cosx Use a graphing calculator to find all solutions to the equation tan = in the 1 cos x interval[0, π ]. Include a sketch of your graphs as well as an explanation of how you reached your answer. 1 (1.5708, 1.) A -point response gives accurate sketches of the two graphs for the given interval and indicates the correct point where the two intersect. A 1-point response has one or more errors in one or both of the two graphs but uses otherwise valid reasoning to create an answer to the equation. A 0-point response shows no mathematical understanding of the problem. Page 48 of 163 Columbus Public Schools 1/5/06

51 Teacher Introduction Trigonometry Periodic Functions Periodic functions are functions where the values repeat themselves at regular intervals. There is a c such that f(t + c) = f(t) for every t in the functions domain, where c is the period length. The length can be found by determining the distance between two maxima (or minima values). If the periodic function is shifted horizontally by its period, in either direction, the resulting graph will be the same as the original graph. The midline of a periodic function is a horizontal line halfway between the minimum and maximum values. The amplitude is the vertical distance, i.e. height difference, between the functions maximum and its midline. For example: F(x) = x int(x) with Δx = 0.1 should be graphed by the student. The student should look not only at the graph, but at the table of values. This function is periodic with a period of 1. Sine and Cosine Functions The sine and cosine functions should be studied using a unit circle with a radius of 1 and centered at the origin. Each point on the circle can be located by its angle of rotation, θ, where y = sin θ and x = cos θ. The pictures of these are included in the strategies for Learning Goal B. Unit Circle (0, 1) (x, y) (1, 0) θ (1, 0) (0, -1) To form triangles, you extend a ray from the center to the side of the circle and draw an altitude to the x-axis. The altitude and x-axis form a right angle and the ray becomes the hypotenuse of the right triangle, with length = 1. The angle θ is the angle the ray makes with the x-axis. The adjacent side is on the x-axis with length equal to the value of the x-coordinate. The opposite side is the altitude with length equal to the value of the y-coordinate. This can be generalized for non-unit circles of radius r: x = rcosθ and y = rsinθ. If θ is the angle measured from the positive x-axis, and P(x, y) is the point on the circle that intercepts the terminal ray, then a right triangle is formed with the hypotenuse as part of the Page 49 of 163 Columbus Public Schools 1/5/06

52 terminal ray. By using the Pythagorean Theorem, cos θ + sin θ = 1 (Pythagorean Identity) can be proved. On the unit circle, we represent angles as rotations of a ray counter-clockwise from the positive x-axis. The x-axis is the initial side of the angle and the ray is the terminal side of the angle. An angle formed in this way is said to be in standard position. Terminal Side (0, 1) (-1, 0) θ (1, 0) (0, -1) Initial Side Example: Use the height of a Ferris wheel by generating the points for 0 o to 360 o for the equation y = 5 + 5sinx and creating a scatterplot to illustrate this concept. Radians 1 radian is the angle at the center of a unit circle which spans an arc of length one. Radians are commonly used in analytical trigonometry and in calculus. The formula for Arc length is: S = rθ. π radians = 180 o Students will need to be familiar with changing the mode on the graphing calculator from degrees to radians and vice versa. Graphs of Sine and Cosine Relate the unit circle to the graphs of the sine and cosine functions with special emphasis on the nd, 3 rd, and 4 th quadrants. When 0 < θ < 90 ; cos θ, sin θ, and tan θ have positive values because x and y are both positive in the first quadrant. When 90 < θ < 180 ; cos θ and tan θ have negative values because x is negative in the second quadrant. Sin θ has a positive value because y is positive in the second quadrant. Page 50 of 163 Columbus Public Schools 1/5/06

53 When 180 < θ < 70 ; cos θ and sin θ have negative values because x and y are negative in the third quadrant. Tan θ has a positive value. When 70 < θ < 360 ; sin θ and tan θ have negative values because y is negative in the fourth quadrant. Cos θ has a positive value because x is positive in the fourth quadrant. If students have trouble remembering the exact values of the sine and cosine functions for 6 π, π π, and, remind them of the relationships in a 30, 60, 90 and a 45, 45, 90 triangle. 4 3 π π o 45 o o π 6 45 o 3 1 π 4 As students compare the two graphs, they should recognize that both have a period of π, an amplitude of 1, a domain of (-, ) and a range of [-1, 1]. They should also note that the sine function is odd while the cosine function is an even. Like other functions, sine and cosine functions can be shifted horizontally, vertically, inverted, compressed, stretched, or a combination of those shifts. Sinusoidal Functions Any transformation of the sine and cosine function is called a sinusoidal function. Students will be working from the general form y = asin b( x+ c) + d where the different variables have the following properties (note this is slightly different than what the book does): a is called the amplitude; it is the distance from the highest point of the sine curve to the mid-line. π b does not have a name itself but is used to determine the period: p =. The period is b how many times the graph completes a cycle in a π interval. d is the vertical shift; it tells how far up or down the entire sine curve is shifted. c gives the phase shift; how far left or right the graph is shifted (a positive c means shift to the left; a negative c means shift to the right). One of the big challenges in analyzing sinusoidal functions is to turn the given function into something that looks like the above form so it can be analyzed easily. For example, the given equation y = 4 + sin(3 x π ) is usually approached by factoring the 3 out, to give something Page 51 of 163 Columbus Public Schools 1/5/06

54 π like y 4 sin3( x ) = + 3, which allows us to see that the period length is 3 is 3 π units to the right. π and the phase shift One thing that will help when students learn to convert equations into this standard form is that they can put both into their graphing calculator. If they are correct, both graphs will wind up on top of each other. Solving Trigonometric Equations Remind students that it is better to remember the Pythagorean Theorem as leg + leg = hypotenuse rather than a + b = c, since there is no guarantee that c is always the hypotenuse. There are two special right triangles. The first is a triangle. The special ratio is 1:1:. The second is a triangle. The special ratio is 1: 3 :. Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the opposite side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. There are many different ways to help your students remember the sine, cosine, and tangent functions. Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief was also a great mathematician. And he developed sine, cosine, and tangent to match his name. SOH (sin = opp / hyp) CAH (cos = adj / hyp) TOA (tan = opp / adj) The following phrase could also be used. Some Old Horse Caught Another Horse Taking Oats Away The angle of elevation is the angle between the line of sight and the horizontal when looking up. The angle of depression is the angle between the line of sight and the horizontal when looking down. It is helpful to remember that the angle of elevation and the angle of depression are alternate interior angles to each other. Look down to person depression elevation Look up to bird Real life applications are architecture and engineering. Page 5 of 163 Columbus Public Schools 1/5/06

55 Non-Right-Angle Trigonometry (Law of Sines/Law of Cosines) We use the Law of Sines and Law of Cosines when we need to solve a triangle that is not a right triangle. By solving, we mean determining the missing side lengths and angle measures, given a minimum of three of these. Students are already familiar with the Pythagorean Theorem which relates the three side-lengths of a right triangle to each other. These two new laws are also about relating information to other information. To use either law, we must use the convention that the three sides of a triangle are labeled a, b, and c, and that their opposite angles are labeled A, B, and C, respectively. The Law of Sines states that for any triangle: sin A sin B sin C = = a b c What we need to know in order to use this law is one side length and its opposite angle. In other words, we must know either A and a, B and b, or C and c. The other piece of information we are given determines which other fraction we will pick out. For example, if we are given the values of C, c, and B, then we will set up the proportion, sin B = sin C, allowing us to find the length of b c side b. We cannot use the Law of Sines if we do not have three pieces of information, and they must meet the requirement we just mentioned. In this last example, if we were given C, c, and b, then we would use the same proportion, but that would leave us with a value for the sin of B, and we would still have to take the inverse sine in order to determine the angle. There are a few situations where the Law of Sines yields unexpected results. First, consider a case where a = 7, b = 8, and A = 100º. Attempting to use the Law of Sines to find the measure of B will give no answer (try it yourself). This is exactly as we expect and want, since in any triangle, the longest side must lie opposite the biggest angle. This cannot be, because there cannot exist any angle bigger than 100º in this triangle, but the 100º is opposite the 7 which isn t the longest side. Another interesting type of result is what is called the ambiguous case. Here, the Law of Sines may give two possible (and viable) results for a missing angle. The Law of Cosines reads: c = a + b abcosc This law relates the three side-lengths to the measure of one angle. The two most common uses of this law are: 1. To find the missing side when the given information is arranged in side angle side formation.. To find a missing angle when all three side-lengths are known. Here, what we are really doing is solving the above equation as if it were in this form: 1 c a b C = cos ab Page 53 of 163 Columbus Public Schools 1/5/06

56 One important property of the Law of Cosines that should be pointed out to students is that when angle C is 90º (when we have a right triangle), then the equation degenerates into the familiar Pythagorean Theorem. In solving a triangle, you will not have to use the Law of Cosines more than once. Once you know all three side-lengths and an angle, you can use the Law of Sines to determine another angle. Also, don t forget such geometry basics as the fact that the angle measures of a triangle must add up to 180º. If you have of them, it s a simple computation to get the third. Trigonometric Identities The section on trigonometric identities should emphasize their use in verifying trigonometric formulas, such as the double angle formula and the reductions rules, and their application in solving trigonometric equations. Although students should have some experience in verifying identities algebraically, this is not the focus of the section. Students should know that, although identities can be verified graphically, this does not constitute a proof. Trigonometric Regression Trigonometric Regression is expected to be done using technology. Please refer to your Graphing Calculator Resource Manual. Page 54 of 163 Columbus Public Schools 1/5/06