Trig Functions PS Sp2016 NAME: SCORE: INSTRUCTIONS PLEASE RESPOND THOUGHTFULLY TO ALL OF THE PROMPTS IN THIS PACKET. TO COMPLETE THE TRIG FUNCTIONS PROBLEM SET, YOU WILL NEED TO: 1. FILL IN THE KNOWLEDGE & COMPREHENSION COMPENDIUM. 2. WORK OUT APPLICATION & ANALYSIS PRACTICE PROBLEMS FROM EUREKA MATH LESSON 1 PROBLEM SET #1-10, LESSON 2 PROBLEM SET #1-8, LESSON 11 PROBLEM SET #1-8, LESSON 13 PROBLEM SET #3-4. 3. WORK OUT SYNTHESIS & EVALUATION PRACTICE PROBLEMS #1-2. FOR EXTRA PRACTICE IN PREPARATION FOR MAJOR ASSESSMENTS, PLEASE CONSIDER DOING ANY OF THE FOLLOWING: 1. KC READINESS ASSESSMENT: WRITING EACH KC COMPENDIUM PROMPT ON ONE SIDE OF A FLASH CARD AND YOUR RESPONSE ON THE OTHER SIDE. THEN, QUIZ YOURSELF NIGHTLY BEFORE GOING TO BED UNTIL YOU RECALL THEM ALL PERFECTLY. 2. AA READINESS ASSESSMENT: WORK OUT SOLUTIONS TO THE ODD- NUMBERED EXERCISES IN SECTION 4.1-4.8, 5.1-5.2 OF YOUR PRECALCULUS TEXTBOOK, AND CHECK YOUR ANSWERS IN THE APPENDIX. AND/OR SIGN UP FOR A FREE ACCOUNT ONLINE AT KHANACADEMY.ORG. CHOOSE THE WORLD OF MATH MISSION, AND ACHIEVE PRACTICED LEVEL 1 ON THE FOLLOWING EXERCISES: 3. SE ASSESSMENT: REDO THE SE PRACTICE PROBLEMS #1-2 MAKING DIFFERENT CHOICES THAN YOU MADE THE FIRST TIME. (SOME A STUDENTS HAVE REPORTED REDOING SE PRACTICE PROBLEMS SEVERAL TIMES TO ACHIEVE SE READINESS.)
Trig Functions PS Sp2016 Directions: Engage your Knowledge and Comprehension (KC) learning during exploration by actively listening for and committing here to a written record of the information that accurately responds to the following prompts. Memorize this information by reviewing it for homework every evening before the KC Assessment. Trigonometric/Inverse Trigonometric Functions 1. Give an example of a trigonometric function with nonzero a, b, c, d transformations, and recognize its parent function. 2. Summarize the four transformations of a parent function arithmetically & graphically. Arithmetically Graphically a b c d 3. State the definition of periodic function, and give an example of this general property from trigonometry. 4. State the reciprocal, quotient, cofunction, and even/odd identities. Reciprocal Quotient Cofunction Even/Odd 5. Recall the equivalence of one angular rotation measured in degrees and in radians. This fact is used to convert any angle from degrees into radians (or from radians into degrees). 6. Illustrate the six trigonometric parent functions over their fundamental periods (in radians), and label their intercepts, extrema, and asymptotes. 7. Paraphrase the definition of inverse relations, and recall the line of symmetry for a pair of inverse relations. 8. Recall the result of composing two functional inverses. 9. Give an example of an inverse trigonometric function with nonzero a, b, c, d transformations, and recognize its parent function. 10. Illustrate the six inverse trigonometric parent functions on their defined ranges, and label their intercepts, extrema, and asymptotes.
Lesson 1 Problem Set 1. Complete the chart below. () () () 2. Evaluate the following trigonometric expressions, and explain how you used the unit circle to determine your answer. a. cos + b. sin c. sin 2 d. cos + e. cos f. cos 2 g. tan + h. tan i. tan 2 3. Rewrite the following trigonometric expressions in an equivalent form using +,, or 2 and evaluate. a. cos b. cos c. sin d. sin e. tan f. tan Lesson 1: Special Triangles and the Unit Circle S.4
Lesson 1 4. Identify the quadrant of the plane that contains the terminal ray of a rotation by if satisfies the given conditions. a. sin() >0 and cos() >0 b. sin() <0 and cos() <0 c. sin() <0 and tan() >0 d. tan() >0 and sin() >0 e. tan() <0 and sin() >0 f. tan() <0 and cos() >0 g. cos() <0 and tan() >0 h. sin() >0 and cos() <0 5. Explain why sin () + cos () =1. 6. Explain how it is possible to have sin() <0, cos() <0, and tan() >0. For which values of between 0 and 2 does this happen? 7. Duncan says that for any real number, tan() = tan(). Is he correct? Explain how you know. 8. Given the following trigonometric functions, identify the quadrant in which the terminal ray of lies in the unit circle shown below. Find the other two trigonometric functions of of sin(), cos(), and tan(). a. sin() = 1 and cos() >0. 2 b. cos() = 1 and sin() >0. 2 c. tan() =1 and cos() <0. d. sin() = 3 and cot() <0. 2 e. tan() = 3 and cos() <0. f. sec() = 2 and sin() <0. g. cot() = 3 and csc() >0. Lesson 1: Special Triangles and the Unit Circle S.5
Lesson 1 9. Toby thinks the following trigonometric equations are true. Use =,, and to develop a conjecture whether or not he is correct in each case below. a. sin() = cos. b. cos() = sin. 10. Toby also thinks the following trigonometric equations are true. Is he correct? Justify your answer. a. sin = sin() sin b. cos 2 = cos() cos 3 c. tan 3 6 = tan 3 tan 6 d. sin + = sin() + sin 6 e. cos + = cos() + cos 4 Lesson 1: Special Triangles and the Unit Circle S.6
Lesson 2 Lesson Summary For all real numbers for which the expressions are defined, sin() =sin(2 + ) and cos() =cos(2 + ) for all integer values of tan() = tan( + ) for all integer values of sin() =sin(), cos() =cos(), and tan() = tan() sin + = cos() and cos + = sin() sin = cos() and cos = sin() Problem Set 1. Evaluate the following trigonometric expressions. Show how you used the unit circle to determine the solution. a. sin b. cos c. tan d. sin e. cos f. sin g. cos h. tan i. sin j. cos k. tan Lesson 2: Properties of Trigonometric Functions S.12
Lesson 2 2. Given each value of below, find a value of with 0 2 so that cos() = cos() and. a. = 3 4 b. = 5 6 c. = 11 12 d. =2 e. = 7 5 f. = 17 30 g. = 8 11 3. Given each value of below, find two values of with 0 2 so that cos() =sin(). a. = 3 b. = 7 6 c. = 3 4 d. = 8 4. Given each value of below, find two values of with 0 2 so that sin() =cos(). a. = 3 b. = 5 6 c. = 7 4 d. = 12 5. Jamal thinks that cos = sin + for any value of. Is he correct? Explain how you know. 6. Shawna thinks that cos = sin + for any value of. Is she correct? Explain how you know. 7. Rochelle looked at Jamal and Shawna s results from Problems 5 and 6 and came up with the conjecture below. Is she correct? Explain how you know. Conjecture: cos() = sin +. Lesson 2: Properties of Trigonometric Functions S.13
Lesson 2 8. A frog is sitting on the edge of a playground carousel with radius 1 meter. The ray through the frog s position and the center of the carousel makes an angle of measure with the horizontal, and his starting coordinates are approximately (0.81,0.59). Find his new coordinates after the carousel rotates by each of the following amounts. a. b. c. 2 d. e. f. g. 2 h. 2 Lesson 2: Properties of Trigonometric Functions S.14
Lesson 11 Problem Set 1. Sketch the graph of = sin() on the same set of axes as the function () = sin(4). Explain the similarities and differences between the two graphs. 2. Sketch the graph of = sin 2 on the same set of axes as the function () = 3sin. Explain the similarities 2 and differences between the two graphs. 3. Indicate the amplitude, frequency, period, phase shift, horizontal and vertical translations, and equation of the midline. Graph the function on the same axes as the graph of the cosine function () = cos(). Graph at least one full period of each function. () = cos 3 4. 4. Sketch the graph of the pairs of functions on the same set of axes: () =sin(4), () = sin(4) +2. 5. The graph and table below show the average monthly high and low temperature for Denver, Colorado. (source: http://www.rssweather.com/climate/colorado/denver/) Write a function to model the average monthly low temperature as a function of the month. Extension: 6. Consider the cosecant function. a. Use technology to help you sketch = csc() for 0 4, 4 4. b. What do you notice about the graph of the function? Compare this to your knowledge of the graph of = sin(). Lesson 11: Revisiting the Graphs of the Trigonometric Functions S.80
Lesson 11 7. Consider the secant function. a. Use technology to help you sketch = sec() for 0 4, 4 4. b. What do you notice about the graph of the function? Compare this to your knowledge of the graph of = cos(). 8. Consider the cotangent function. a. Use technology to help you sketch = cot() for 0 2, 4 4. b. What do you notice about the graph of the function? Compare this to your knowledge of the graph of = tan(). Lesson 11: Revisiting the Graphs of the Trigonometric Functions S.81
Lesson 13 Problem Set 1. Consider the situation of sitting down with eye level at 46 in.. Find the missing distances and heights for the following: a. The bottom of the picture is at 50 in. and the top is at 74 in. What is the optimal viewing distance? b. The bottom of the picture is at 52 in. and the top is at 60 in. What is the optimal viewing distance? c. The bottom of the picture is at 48 in. and the top is at 64 in. What is the optimal viewing distance? d. What is the height of the picture if the optimal viewing distance is 1 ft. and the bottom of the picture is hung at 47 in.? 2. Consider the situation where you are looking at a painting inches above your line of sight and inches below your line of sight. a. Find the optimal viewing distance if it exists. b. If the average standing eye height of Americans is 61.4 in., at what height should paintings and other works of art be hung? 3. The amount of daylight per day is periodic with respect to the day of the year. The function = 3.016 cos 2 + 12.25 gives the number of hours of daylight in New York,, as a function of the number of 365 days since the winter solstice (December 22), which is represented by. a. On what days will the following hours of sunlight occur? i. 15 hours, 15 minutes. ii. 12 hours. iii. 9 hours, 15 minutes. iv. 10 hours. v. 9 hours. b. Give a function that will give the day of the year from the solstice as a function of the hours of daylight. c. What is the domain of the function you gave in part (b)? d. What does the domain tell you in the context of the problem? e. What is the range of the function? Does this make sense in the context of the problem? Explain. 4. Ocean tides are an example of periodic behavior. At a particular harbor, data was collected over the course of 24 hours to create the following model: = 1.236 sin + 1.798, which gives the water level,, in feet above 3 the MLLW (mean lower low water) as a function of the time,, in hours. a. How many periods are there each day? b. Write a function that gives the time in hours as a function of the water level. How many other times per day will have the same water levels as those given by the function? Lesson 13: Modeling with Inverse Trigonometric Functions S.93
Trig Functions SE Practice Sp2016 1. Construct the rule for a trigonometric function f that has the following properties, and evaluate how analytically: a. is invertible on [ 5π 2, 3π 2 ] b. lim f x x 0 + 1 ( ) = + c. is a transformation of an even parent function d. simplifies to an odd parent function. Then, parametricize f using a nontrivial parameter, and show how an establishment of its orientation by limits will not work but by evaluation will. 2. Construct the rule for a trigonometric function f that has the following properties, and evaluate how analytically: a. is invertible on [ π 2, 3π 2 ] b. lim f x x 0 1 ( ) = c. is a transformation of an odd parent function d. simplifies to an odd parent function. Then, parametricize f using a nontrivial parameter, and show how an establishment of its orientation by limits will not work but by evaluation will.