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Integrated Math 2 Unit 5 Right Triangle and Circular Trigonometry Farmington Public Schools Grade 9-11 Math Hall, Lepi Draft: 6/27/2003 Farmington Public Schools Table of Contents Unit Summary ………………….….…..page - 3 Stage One: Standards Stage One identifies the desired results of the unit including the broad understandings, the unit outcome statement and essential questions that focus the unit, and the necessary knowledge and skills. The Understanding by Design Handbook, 1999 …………………………….... page(s) 4-6 Stage Two: Assessment Package Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge and skills identified in Stage One. ……………………………… page 7 - 8 Stage Three: Curriculum and Instruction Stage Three helps teachers plan learning experiences and instruction that aligns with Stage One and enables students to be successful in Stage two. Planning and lesson options are given, however teachers are encouraged to customize this stage to their own students, maintaining alignment with Stages One and Two. ………………..……………… page 9 - 10 Appendices ….....………………………. page 11- 59 Hall, Lepi Draft 6/27/2003 Farmington Public Schools 2 Unit Summary This unit on Right Triangle and Circular Trigonometry is the fifth unit of the year following a unit on Power Models. It addresses the Operations, Estimation and Approximation, Ratios Proportions and Percents, Measurement and Algebra and Functions. The unit builds on previous knowledge of transformations, proportional reasoning, characteristics of right triangles and basic knowledge of circles. It explores trigonometric functions graphically, algebraically and numerically. Applications include architecture, physics, and earth science. This is the first formal introduction of Trigonometric topics. This unit will take between five and six weeks. Hall, Lepi Draft 6/27/2003 Farmington Public Schools 3 Stage One: Standards Stage One identifies the desired results of the unit including the broad understandings, the unit outcome statement and essential questions that focus the unit, and the necessary knowledge and skills. The Understanding by Design Handbook, 1999 Essential Understandings and Content Standards Essential Math Understanding 2 Operations Students will understand that people must correctly select and apply appropriate number operations in order to solve numerical problems 3 Estimation and Approximation Students will understand that people must use estimation and approximation in order to judge the reasonableness of results and to guide their mathematical thinking 4 Ratios, Proportions and Percents Students will understand that people use ratios, proportions and percents in order to represent relationships between quantities and measures. 5 Measurement Students will understand that people must appropriately apply customary and metric measurement units in order to approximate, measure and compute length, area, volume, mass, temperature, angle and time 9 Algebra and Functions Students will understand that people must use algebraic skills and concepts, including functions, in order to describe real-world phenomena symbolically and graphically, and to model quantitative change. Hall, Lepi Draft 6/27/2003 Content Standard The students will be able to: c. Use appropriate methods for computing, including mental math, estimation, paper-andpencil and calculator methods a. Assess the reasonableness of answers to problems arrived at using paper-and-pencil techniques, mental math, formulas, calculators and computers a. Use ratios, proportions… to solve a wide variety of situations, including real world problems d. Describe trigonometric ratios and apply them to measuring triangles d. Use techniques of algebra, geometry and trigonometry to measure quantities indirectly b. Model real-world phenomena using… trigonometric functions d. Translate among and use tabular, symbolic and graphical representations of equations…and functions Farmington Public Schools 4 Unit Outcome Statement Consistently aligning all instruction with this statement will maintain focus in this unit. This unit will introduce students to trigonometric (a.k.a. periodic) functions and their uses. Right triangle trigonometry will focus students’ attention on how to use trigonometry to solve problems involving right triangles and the relationship between angle of elevation, angle of depression, and line of sight in solving such problems. Circular trigonometry will focus students’ attention on what periodic change is, how to model periodic change graphically and symbolically, the relationship between area and circumference of a circle and its parts (sectors, arcs) and how to measure angles in both degrees and radians. The students will: Define sine, cosine and tangent for an acute angle of a right triangle. Use the trigonometric ratios of sine, cosine and tangent to find missing sides and angles in right triangles. Graph the sine and cosine function to model periodic motion. Recognize and perform transformations of the sine and cosine, both graphically and symbolically using the form y = a sin (bx) +d or y = a cos (bx) +d. Convert angle measurements between degrees and radians. Find area, circumference, arc length and area of a sector for circles. Essential Questions These questions help to focus the unit and guide inquiry. What is trigonometry? How is trigonometry used to model and solve real world problems and situations? How does trigonometry relate to the geometric models of triangles and circles? Hall, Lepi Draft 6/27/2003 Farmington Public Schools 5 Knowledge and Skills The Knowledge and Skills section includes the key facts, concepts, principles, skills, and processes called for by the content standards and needed by students to reach desired understandings. The Understanding by Design Handbook, 1999 Knowledge Students will: Identify the parts of the sine and cosine models: y= d + a sin(bx) and y= d + a cos (bx). Define the sine, cosine and tangent of an acute angle of a right triangle. Be able to accurately draw a model using line of sight, angle of depression and angle of elevation. Know when to use formulas for area, circumference, arc length and sectors of circles. Be able to distinguish between the graphs of sine and cosine. Skills/Processes Students will: Find the value for the sine, cosine and tangent of an acute angle. Convert a radian measurement to degrees, and vice versa. Use appropriate trigonometric functions to find the missing side or angle of a right triangle. Find the arc length, area, circumference, and area of a sector of a circle. Graph accurately the sine and cosine functions. Thinking Skills Students will: Understand and apply how changes in a, b and d affect the graph of the trigonometric functions (in isolation and combination). Understand the connection between the transformations used in the geometry unit taught earlier and trigonometric graphs. Understand that all similar right triangles will have the same values for their trigonometric ratios. Understand how the sine and cosine functions model periodic behavior of real world applications. Hall, Lepi Draft 6/27/2003 Farmington Public Schools 6 Stage Two: Assessment Package Stage Two determines the acceptable evidence that students have acquired the understandings, knowledge and skills identified in Stage One. Authentic Performance Task Triangle Trig Project Students will use the methods learned, similar triangle proportions and right triangle trigonometry ratios to find the height of two “tall” structures on the high school campus. They will describe the methods used, show all calculations, and give the pros and cons of the methods used for each object that they are estimating the height of. This will be turned in as a written report of their findings to preclude a job. Tests, Quizzes, and Other Quick and Ongoing Checks for Understanding Quiz 1 Trig Ratio Quiz On this assessment students will use the trigonometric ratios for sine, cosine, and tangent to identify the length of sides of triangles, both in fraction and decimal form. Students will also find the measure of an angle using inverse trig ratios. Quiz 2 Trigonometry Missing sides Quiz On this assessment students will identify and use an appropriate method for calculating the missing sides of right triangles. Test 1 Right Triangle Trigonometry On this test for right triangle trigonometry students will identify the parts of a right triangle using a particular angle as reference, use the trig ratio or its inverse appropriately to find lengths of sides or angles of a right triangle, translate word problems into appropriate diagrams in order to solve for the missing parts. Quiz 3 Trig graph quiz On this non-calculator quiz for sine and cosine graphs, students will use their knowledge of amplitude, period and vertical shift in order to match a graph with its equation. Test 2 Graphs of sine and cosine On this test, students will be allowed to use a calculator to accurately graph different sine and cosine functions, while paying attention to period, amplitude and vertical shifts, as well as identify these components when given an equation or graph, a word problem will be solved that shows the students have made the connection between degrees and revolutions as well as how to graph the information and come up with an appropriate equation for the situation. Hall, Lepi Draft 6/27/2003 Farmington Public Schools 7 Test 3 Circle Test On this test, students will use their knowledge or circles to find area, circumference, arc length and area of a sector. They will convert between degree and radian measurement, graph a trig function using radian measurement and solve word problems involving circles. Projects, Reports, Etc. Circle Project In this project each student will be assigned one of the satellites for one of the planet in our solar system to investigate the concept of circumference and arc length as it relates to satellite orbits. Hall, Lepi Draft 6/27/2003 Farmington Public Schools 8 Stage Three: Learning Experiences and Instruction Stage Three helps teachers plan learning experiences and instruction that align with Stage One and enables students to be successful in Stage Two. Learning Experiences and Instruction The learning experiences and instruction described in this section provide teachers with one option for meeting the standards listed in Stage One. Teachers are encouraged to design their own learning experiences and instruction, tailored to the needs of their particular students. High School Lesson Topic Right Triangle Trigonometry Applications of right triangle trig Guiding Questions Why does knowing the measure of one acute angle of right triangle determine the triangle shape? What are the trig ratios? What can you say about the ratio of corresponding sides of similar right triangles? How do the trigonometric ratios aid in solving for missing sides of right triangles? How do inverse trigonometric ratios aid in solving for missing angles? How are trigonometric ratios used to solve real world problems? Explain how are angle of depression, angle of elevation and line of sight related. Suggested Sequence of Teaching and Learning Activities Investigation 2.2 Pg. 395-399 OYO Pg. 399 Sine, Cosine, and Tangent worksheets Pg. 408 O1, R2 Approx. 4 days Investigation 2.3 Pg. 400- 405 OYO Pg. 405 Word Problem worksheet Find missing length worksheet Pg. 406 M1, M2, M3, O5 Authentic Assessment Trig Problem Set trig problems Approx. 6 days Sine and cosine graphs Explain how trigonometry can be used to model aspects of periodic motion. Describe the pattern of change for a basic sine and cosine graph. How are these trig graphs different from other graph models we have investigated this year? Explain how the amplitude, period and vertical shift of a trig function relates to the graphs of sine and cosine. How are each of these pieces represented in the equation? Paper Plate Activity Investigation 3.3 Pg. 431-434 OYO pg. 435 Ferris Wheel Problem Unit Circle and Sine/cosine worksheet OYO pg. 441 Graphing Sine/Cosine worksheet Amplitude and Vertical Shift worksheet Trig graphs and matching equations Pg. 442 M1, o3, o4, R1, Pg. 452 #5 a Approx. 8 days Hall, Lepi Draft 6/27/2003 Farmington Public Schools 9 Circles circumference and area, arc length and area sectors What is the relationship between arc length and circumference? Area of circle and area of sector? How can arc length and area of a sector be modeled in real world situations? How is central angle, arc length and sector area related? Circumference, Area and Sector worksheet Circumference and Arc Length worksheet Arc, Sectors etc. worksheet Approx. 3 days Radian and degrees What is a radian? How are revolutions, radian measurement and degrees related? Explain how to change radian measurement to degrees and vice versa. Investigation 3.2 Pg. 419-422 # 1-6 Investigation 3.4 pg. 436-44 Pg. 446 R2 Revolutions Worksheet Radian/Degree conversion worksheet Approx. 5 days Hall, Lepi Draft 6/27/2003 Farmington Public Schools 10 Appendices Tests, quizzes, authentic assessment, and project attached. Worksheets attached. Hall, Lepi Draft 6/27/2003 Farmington Public Schools 11 Integrated Math 2 Trig Ratio Quiz Name______________________________ Date _________ Use the diagram to the right to answer the following questions. 1. Measure of < C _________ 2. Measure of < A _________ 3. Length of side opposite < A ______ 4. Length of side adjacent to < B 5. Length of side adjacent to < A ________ 6. Length of hypotenuse _________ 11 ‘ 5’ ________ 27 9.8’ Use ∆ DEF to find the following trigonometric ratios, then give the decimal value for the trig ratio accurate to thousandths, (3 decimal places). 9 7. Sin < D ______ _______ 8. Cos < D ______ _______ 6m 9. Tan < D ______ _______ 10.8 m 10. Sin < F ______ _______ 11. Cos < F ______ ______ 12. Tan < F _____ _______ Use the Pythagorean theorem to determine the missing side length (to the nearest tenth). Then find the following trigonometric ratios, convert to decimal form to nearest thousandths. 1. 13. Sin < R _____ ______ 14. Cos < R______ ______ 18 ft 15. Tan < R ______ ______ 16. Sin < P ______ ______ M 17. Cos < P _____ ______ P 17 ft 18. Tan < P _____ ______ Hall, Lepi Draft 6/27/2003 Farmington Public Schools 12 For each of the trig ratios below fill in the correct angle name. K 4 J 3 5 H 19. 3 = sin ∠ ______ 5 Hall, Lepi 20. 4 = cos ∠ ______ 5 Draft 6/27/2003 4 21. = tan ∠ ____ 3 Farmington Public Schools 13 Integrated Math 2 Trigonometry- Missing Sides Quiz Name_________________________ Date __________________ Identify the trig function to be used to solve for x; include the set up and work! 1. 4 x X = ____________ 39º 2. 10 20º X = ____________ x 3. 27 14º X = ____________ x Hall, Lepi Draft 6/27/2003 Farmington Public Schools 14 27º 4. x 7 X = ____________ 5. 12 x X = ____________ 57º 10 6. Given ∆ABC with right angle C, draw and label the triangle then find all the missing parts. ∠A = 73° , c = 109 cm Hall, Lepi Draft 6/27/2003 Farmington Public Schools 15 Integrated Math 2 Triangle Trig Assessment Name ______________________ Date _________ You have applied for a job with RJ Construction as a supervisor of construction. During the interview, a discussion involving finding indirect measurements of tall structures included both similar triangle measurement without knowing the angle measures and the use of trigonometric ratios where knowing the angle is needed. The interviewer Mr. Hi was interested in your ability to measure tall structures in more than one way in order to ensure the accuracy of calculation. In order to get this job, you have agreed to complete the following report for him showing your ability to use two different methods of finding the height of a tall structure. The report is due within 5 days (school days) and the directions for this report are as follows. Directions: 1. Describe the two different methods of measuring the height of tall structures. One method should use similar triangles, without knowing the angle measures, and the other method should trigonometry, using the angle of elevation. (One method using shadows on a sunny day, or mirrors.) 2. Use both methods to measure the height of two structures on the FHS campus. (Ex. Flag pole, backstop on baseball field, height of building etc.) 3. In your report, identify and describe the structures whose height you measure. For each structure and method of measurement, describe and give the step-bystep results of the measurements you made. Describe the mathematical calculations you preformed to get the final results. 4. Compare the numerical results of the two measurement methods. Discuss the possible sources of inaccuracy in each method, the relative difficulty of applying each method, the conditions under which each method works best, and the kind of structures that are measured accurately using each method. Grading Rubric: A. Format of the report follows directions B. Clarity and accuracy of the descriptions of measurement methods. C. Comparison of measurement results D. Quality of discussion of pros and cons of measurement methods. Hall, Lepi Draft 6/27/2003 10% 35% 20% 35% Farmington Public Schools 16 Integrated Math 2 Test 1, Unit 6 Trigonometric Ratios Name ______________________ Round all answers correctly to one decimal place! 1. Relative to angle P, label the 3 sides of the triangle shown to the right with the terms “opposite”, “hypotenuse”, and “adjacent”. P 2. Complete the following definition (circle the correct response): In a right triangle, the tangent of an angle is defined as a) the length of the side adjacent to the angle divided by the length of the side opposite the angle. b) the length of the hypotenuse divided by the length of the side adjacent to the angle. c) the length of the side opposite the angle divided by the length of the hypotenuse. d) the length of the side adjacent to the angle divided by the length of the hypotenuse. e) the length of the side opposite the angle divided by the length of the side adjacent to the angle. Use the trigonometric table or your calculator to identify the following (round answers to hundredths) 3. Sin 27 = ____________ tan 53 = ___________ cos 45 = ___________ 4. the sine of the angle is .9965, the angle = ___________ the cosine of the angle is .0456, the angle = __________ the tangent of the angle 1.576, the angle = _________ In questions 5 – 9 round final answers to 2 decimal places and show all work! (no work = no credit) 5. Find the length of x. x = ________________ 6. Find the length of x. Hall, Lepi x 5 25° Draft 6/27/2003 Farmington Public Schools 17 x = ________________ x 8.5 50° 7. Find the length of x. x 10 x = ________________ 40° 8. Find the measure of angle A. A = _______________ A 25 12 9a. A right triangle uses the standard naming conventions with angles A, B, C and side lengths a, b, c. If ∠B = 30° and c = 80 cm, find the measures of side BC two ways (one using trigonometry and one using properties of “special” triangles). Show your work. b. Did you get the same answer using each method? Why do you think this happened? Hall, Lepi Draft 6/27/2003 Farmington Public Schools 18 10. A 15-meter cable is used to support a tree. The wire is staked to the ground and makes a 50° angle with the ground. How high up the tree is the cable connected? Draw a sketch and show all work. 11. A man climbs to the top of a 45-foot flagpole. He looks down at his friend on the ground at an angle of depression of 35°. How far is his friend from the base of the flagpole? Draw a sketch and show all work. 12. If a man with a height of 6 feet casts a 3.5-foot shadow on the ground, what angle are the suns rays making with the ground? Draw a sketch and show all work. Bonus: Karen is 1.6 meters tall looking at the top of a flagpole that is 10 meters away. The angle of elevation is 42 ° . Find the height of the flagpole to the nearest tenth. Integrated Math 2 Hall, Lepi Name _________________ Draft 6/27/2003 Farmington Public Schools 19 Trig graph quiz NO CALCULATOR Date ________ Match each of the following graphs with one of the equations listed. All windows are 0 – 360 on the x axis and –5 to 5 on the y axis. a. y = sin x b. y = 3 sin x c. y = 2 + cos x e. y = cos 2 x g. y = −1cos x d . y = sin3 x f . y = 3 + sin x h. y = 2cos x Hall, Lepi _______ _________ _______ _______ ________ _________ _______ _________ Draft 6/27/2003 Farmington Public Schools 20 Integrated Math 2 Test, Graphs of Sine and Cosine (with Calculator) Name ________________ 1. Fill in the appropriate values for each function given below. Trig Function y = cos(3x) y = 5sin(x) +1 y = 2cos(4x) y = sin (2x) +3 y = cos(½x) -2 y = -3sin(5x)-1 Amplitude Period (in degrees) Vertical Shift Problems 2-5. Accurately graph the following trig functions. Include several points. 2. Graph y = sin(x) + 2 3. Graph y = –3cos(x) 4. Graph y = cos(2x) +1 5. Graph y = 2sin(x) – 1 6. What is the amplitude of y = 3 + 2sin(4x)? ______________ 7. What is the period of y = 3 + 2sin(4x)? _______________ Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 21 8 – 11, write equations that fit each of the graphs shown. 8. equation: 9. equation: 10. equation: 11. equation: Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 22 12. A carnival ride, called The Wheel of Horror, is like a Ferris wheel that is inside a haunted house. You board the ride from a platform at the height of the center the wheel. From your perspective, the Wheel of Horror turns in a counterclockwise direction. When your seat is above the platform level, you are in the “belfry” where you are bombarded with flying monsters. When your seat is below the level of the platform, you are in the “dungeon” where equally scary creatures lurk. a. The Wheel of Horror has a radius of 5 meters. Sketch The Wheel of Horror, being sure to mark the locations of the platform, the belfry, and the dungeon. b. You and a friend board the Wheel of Horror. Sketch a graph showing your distance, y, from the level of the platform during 2 revolutions of the wheel (where x is in degrees). Mark the scale you use on the y axis. Write an equation modeling this periodic motion. EQUATION: y =_________________________________________ c. Indicate whether you are in the belfry or in the dungeon in each of the following intervals on the x-axis by marking a 3in the appropriate place. (0,180) (180, 360) (360, 540) (540, 720) Belfry Dungeon Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 23 Integrated Math 2 Circle Project Name ____________________ Date __________ In the table below is the information for the radius, distance from the planet and the number of days for the orbital period in days for the path of different satellites/planets in our solar system. Your task is to determine: the length in km. of the path a person would walk on the planet for a specific angle the length of the orbit of the satellite for that same angle how long it takes the satellite to travel that arc. You are to do each of the above for: a 90-degree rotation/movement a 200-degree rotation/movement a 315-degree rotation/movement. In your solution be sure to discuss if these measurements are related and if so how?. Satellite Earth Mars Jupiter Saturn Uranus Neptune Radius km Distance km Orbital period day Moon 1738 384000 27.3 Phobos 12 9370 .32 Deimos 8 23500 1.26 Almalthea 135 181000 .498 Io 1820 421600 1.77 Europa 1560 670900 3.55 Ganymede 2640 1070000 7.16 Calisto 2420 1880000 16.69 Himalia 85 11470000 250.6 Lysithia 12 11710000 260 Mimas 196 186000 .94 Euceladus 250 238000 1.37 Tethys 500 294700 1.888 Dione 560 377000 2.74 Rhea 765 527000 4.52 Titan 2575 1222000 15.95 Miranda 150 130000 1.41 Ariel 665 192000 2.52 Triton 1600 354000 5.88 Grading Rubric: A. Format follows directions B. Correct formulas used in determining responses C. Clarity and accuracy of responses D. Comparison of measurement results Hall, Lepi DRAFT: 06/24/2003 Eccentricity .055 .015 .001 .003 0 0 .001 .01 .158 .13 .02 .004 0 .002 .001 .029 .017 .003 0 10% 35% 35% 20% Farmington Public Schools 24 Integrated Math 2 Circle Test Name ___________________ Date ___________ 1. If a circle has a diameter of 21 meters, then what is the radius of that circle? _______ 2. What is the formula for the circumference of a circle? ___________ 3. The distance from the center of the circle to a point on the circle is called the _______ 4. In terms of π what is the circumference of a circle with radius 9 inches? ________ 5. In terms of π what is the area of a garden with a diameter of 18 feet? ___________ 6. What fraction of a circle does a central angel of 54 degrees cut? _________ 7. How many degrees are in a complete circle? __________ 8. How many radians are in a complete circle? __________ 9. For the given circle find each of there values: a. Radius 40 cm b. Diameter _______ c. Circumference _______ d. Arc length _______ e. Area of circle _______ f. Area of sector _______ 135 10. For the given circle find each of these values: a. Radius _______ b. Diameter 12 inches c. Circumference _______ d. Arc length _______ e. Area of circle _______ f. Area of sector _______ 300 11. What simplified fraction of the circle is shown? _________ 285 12. Convert each degree measurement to radian measurement in fraction form with correct units. Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 25 a. 1200 b. 1800 c. 300 d . 3300 e. 450 f . 2700 13. Convert each radian measurement to degrees. π 2 π 3 b. 2π c. 5 d. π 6 7 e. π 4 f. a. 2 π 12 14. A large pizza has a diameter of 16 inches, and it is cut into 10 equal triangular slices. Half of the pizza is topped with pepperoni and the other half has anchovies. Kate hates anchovies. a. What is the area of one slice of Kate’s pizza? (show all formulas used and clearly state the answer in both exact and approximate form.) b. What is the length of the crust on her piece of pizza? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 26 15. The wheel of a bicycle has a radius of 16 inches. How many inches will the bicycle wheel travel in one complete revolution? (give both exact and approximate answers) Bonus: Graph y = sin x using radian measurements on the x axis, give the period and amplitude of your graph.. Period ___________ Amplitude _____________ π π 2π 2 2 Hall, Lepi 3π DRAFT: 06/24/2003 Farmington Public Schools 27 Integrated Math II Radian and Degree Conversions Name ____________________ A radian measures the length of an arc in relation to the radius of a circle. If you have a circle with a radius of 1, what is the circumference? __________ The circumference of a circle is the arc length of one revolution of the circle. So one revolution of a circle with radius of 1, measures 2π radians. How many degrees are there in one revolution of a circle? __________ It is possible to convert degrees to radians and vice versa, using the fact that 2π radians = 360 degrees so… π radians = _____ degrees Example 1: Convert 135° to radian measure. Simplify answer and leave in terms of π. 135° 180° To convert to radians, set up a proportion: = x π Solve for x: x 135π cross-multiply: 135π = 180x ⇒ = 180 180 135π 3π isolate x by dividing both sides by 180: x = ⇒ x= radians 180 4 Example 2: Convert −3π to degree measure. 5 ⎛ −3π ⎞ ⎜ ⎟ π 5 ⎠ ⎝ Set up a proportion: = x 180° Solve for x: ⎛ −3π ⎞ cross-multiply, then simplify: π x = ⎜ ⎟ 180 ⇒ π x = −108π ⎝ 5 ⎠ π x −108π = ⇒ x = −108° isolate x by dividing both sides by π: π Hall, Lepi DRAFT: 06/24/2003 π Farmington Public Schools 28 Convert each degree measure to radian measure. Simplify answers and leave in terms of π. 1.300° 2. 36° 3. 6° 4.105° 5. -85° 6. 70° 7.75° 8. -860° 9. 1200° Convert each radian measure to degree measure. 5π 11π 10. radians 11. radians 2 12 13. 16. 13π radians 12 π 5 Hall, Lepi radians 14. 4π radians 17. 5π radians 4 DRAFT: 06/24/2003 12. 7π radians 9 15. −12π radians 5 18. 3 radians Farmington Public Schools 29 Integrated Math II Name ____________________ Revolutions One revolution of a circle is equivalent to one rotation. Therefore, there are _____ degrees in one revolution and _____radians in one revolution. Example 1 How many degrees are swept through when a tire rotates Step 1: Set up a proportion: x = 360° 1rev. 3 revolution? 4 3 rev. 4 ⎛ 3⎞ Step 2: Solve for x: 1x = ⎜ ⎟ 360 ⇒ x = 270° ⎝4⎠ Example 2 How many revolutions are swept through when a wheel rotates 480°? x 1rev. = 480° 360° 360 x 480(1) = Step 2: Solve for x: 360 x = 480(1) ⇒ 360 360 3 x= revolution 4 Step 1: Set up a proportion: Example 3 If the radius of a car tire measures 1 foot and the car travels 30 feet, how many revolutions did the tire make? Round answer to the nearest hundredth. Step 1: Find the circumference of the tire: C = 2π(1) ⇒ C = 2π feet x 1rev. = 30 ft 2π ft 2π ( x ) 1(30) Step 3: Solve for x: 2π ( x ) = 1(30) ⇒ = ⇒ x = 4.77 revolutions 2π 2π Step 2: Set up a proportion: I. Practice 1. How many degrees are swept through when a tire rotates 1½ revolutions? 2. How many radians are swept through when a tire rotates 3¾ revolutions? 3. How many revolutions are swept through when a wheel rotates 216°? 4. How many revolutions are swept through when a wheel rotations 5π radians? 5. If the radius of a tractor-trailer tire measures 1.5 feet and the tractor-trailer travels 150 feet, how many revolutions did the tire make? Round answer to the nearest hundredth. Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 30 Problem Set, Trig Problems Name_________________________ Do problems: ___________________________________ and __________________ Directions: Include a diagram for each problem and SHOW ALL WORK! Round all answers to the nearest hundredth (2 decimal places) and include units. 1. 2. Sam elevates his telescope 55° to spot the top of a building from a point on the ground 200 feet away. a. What trigonometric function of the 55° angle is equal to b. Use this function to find the height of the building. A car traveled along a ramp raised 4° from the ground. a. If the car traveled 750 meters along the ramp, how far did it rise during this distance? b. 3. h ? 200 If the car rises 10 meters, how far along the ramp did the car travel? A 150-foot rope is tied from the top of a 100-foot-tall pole to a stake in the ground. a. What is the angle between the rope and the ground? b. How far from the base of the pole is the stake? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 31 4. The Aerial run in Snowbird, Utah, is 8,395 feet long. Its vertical drop is 2900 feet. If the slope were constant, estimate the angle of elevation that the run makes with the horizontal. 5. a. b. 6. A plane at a height of 5000 feet begins descending in a straight line toward a runway. If the horizontal distance between the plane and the runway is 45,000 feet. What angle of depression must the plane use? If the angle of depression from the plane to the runway is 5 degrees, what horizontal distance will the plane travel before it hits the runway? You lean a ladder 6.7 meters long against the wall. a. If the ladder makes an angle of 63° with the ground, how high up is the top of the ladder? b. If the ladder makes an angle of elevation of 70° with the ground, how far is the base of the ladder from the base of the wall? 7. You must order a new rope for the flagpole. To find out what length of rope is needed, you observe that the pole casts a shadow 11.6 meters long on the ground. The angle of elevation of the sun is 36°. How tall is the pole? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 32 8. Your cat is trapped on a tree branch 6.5 meters above the ground. Your ladder is only 6.7 meters long. If you place the ladder’s tip on the branch, what angle will the ladder make with the ground? 9. The tallest freestanding structure in the world is the 553-meter tall CN Tower in Toronto, Ontario. a. Suppose that at a certain time of day it casts a shadow 1100 meters long on the ground. What is the angle of elevation of the sun at that time of day? b. At around 11:00 AM, the angle of elevation of the sun is 75°. What is the length of the shadow cast by the tower? 10. A boat is anchored to the bottom of the sea with a 30-foot rope. a. If a duck, 8 feet from the boat is swimming above the anchor, what is the angle of depression from the boat to the anchor? b. If the angle of elevation from the anchor to the boat is 25°. What is the horizontal distance from the boat to the anchor? 11. The top of a storage space is 4 meters above the main roof of a house. If the main roof is 9.1 meters above the ground and a ladder that reaches the top of the storage space makes an able of 69° with the ground, how long is the ladder? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 33 12. Trevor is standing on top of a cliff 200 feet above a lake a. If the measurement of the angle of depression to a boat on the lake is 21°. How far is the boat from the base of the cliff? b. If a person in the boat is 100 feet from the cliff, what is the angle of elevation to the person at the top of the cliff? c. If Trevor notices a hot air balloon that is 200 feet away horizontally, at an angle of elevation of 30°, how high above the lake is the plane? Extra Credit: John views the top of a water tower at an angle of elevation of 36°. He walks 120 meters in a straight line toward the tower. Then he sights the top of the tower at an angle of elevation of 51°. How far is John from the base of the tower? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 34 Integrated Math 2 Name __________________ Problem set Pg 406 M1) a) Date ________________ A C 63 | | | | | | | 123 m b) river c) B distance M2) - - - - - - - - - - - - - - - - - - - - - - - angle of descent a) altitude Sketch this for each of a, b and c with appropriate values. b) c) d) M3) a) b) depth of crater c) angle of elevation length of shadow Hall, Lepi DRAFT: 06/24/2003 d) Farmington Public Schools 35 Integrated Math 2 Pg. 408-10 Name _______________________ O1) a) B b) sin A = sin A’ = tan A = tan A’ = a c C b A c) R2) Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 36 Integrated Math II Graphing Sine and Cosine Name ____________________ Sketch each pair of functions on the same set of axes. Use 0° ≤ x ≤ 360° . 1. y = cos x , y = 4 cos x 2. 4 1 y = sin x , y = − sin x 2 2 3 2 1 1 90 180 270 360 90 -1 -2 180 270 360 -1 -3 -4 -2 3. y = sin x , y = −3 + sin x 4. y = cos x , y = cos 4 x 4 4 3 3 2 2 1 1 90 180 270 360 90 -1 -1 -2 -2 -3 -3 -4 -4 5. y = cos x , y = 1 + cos x 6. 2 1 1 180 270 360 90 -1 -1 -2 -2 Hall, Lepi DRAFT: 06/24/2003 270 360 1 y = sin x , y = sin x 3 2 90 180 Farmington Public Schools 180 270 360 37 7. y = cos x , y = 2 cos x y = sin x , y = 2 + sin x 8. 4 2 3 2 1 1 90 180 270 90 360 180 270 360 -1 -2 -1 -3 -4 -2 y = cos x , y = cos 2 x 9. 10. y = sin x , y = −1 + sin x 2 2 1 1 90 90 180 270 180 270 360 360 -1 -1 -2 -2 ⎛1 ⎞ 11. y = cos x , y = cos ⎜ x ⎟ ⎝3 ⎠ 12. y = sin x , y = 2 − sin x 4 4 3 3 2 2 1 1 90 180 270 360 90 -1 -1 -2 -2 -3 -3 -4 -4 Hall, Lepi DRAFT: 06/24/2003 180 Farmington Public Schools 270 360 38 Integrated Math II Practice with Circumference and Arc Lengths Find the circumference (in terms of 1. radius = 4 m Name ________________________ ) of the following circles. 2. r = 15 cm 3. The arc of a circle is shown to the right: 6 in. ∩ A 4. Find these arc lengths (express in terms of ¼ of a circle with radius 8 meters = Arc AB ) B ½ of a circle with radius 6 inches = ¾ of a circle with radius 12 yards = ⅔ of a circle with radius 6 cm. = ⅛ of a circle with radius 24 feet = ⅜ of a circle with radius 4 meters = ⅓ of a circle with radius 1 foot = A central angle is an angle with its vertex at the center of a circle. 5. What fraction (simplified) of the circle is each of these arcs that are determined by the central angles given? 120 60 90 Hall, Lepi DRAFT: 06/24/2003 30 Farmington Public Schools 39 6. What fraction (simplified) of the circle is each of these arcs that are determined by the central angles given? 180º 270º 200º 36º 90 40º 45º 300º 7. Find the lengths of the arcs of these circles given these central angles and these radii. Express your answers in terms of Central angle Radius 60º 12 inches 90º 8 feet 120º 15 meters 20º 18 centimeters 300º 30 yards 45º 1 120º 1 180º 1 420º 1 540º 1 Hall, Lepi DRAFT: 06/24/2003 . Arc Length Farmington Public Schools 40 Integrated Math II Trig Ratio Practice Name ________________________ B A. A diagram of ∆ABC is given to the right. Give the measures of the missing sides or angles. 13 5 1. ∠C 2. ∠A 3. Side opposite ∠ A 4. Side (leg) adjacent to ∠ B 5. Side (leg) adjacent to ∠ A 6. Hypotenuse C 12 A For ∆DEF as shown below, find the following trigonometric ratios…then divide and give the II. ratio to the nearest thousandth (3 decimal places). 24 F E Sin ∠ D = __________ = 7 Cos ∠ D = __________ = 25 Tan ∠ D = __________ = D Sin ∠ F = __________ = Cos ∠ F = _________ = Tan ∠F = _________ = Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 41 III.Use the Pythagorean Theorem to determine the missing side length (to the nearest tenth). Then find the following trigonometric ratios. Then divide and give the ratio to the nearest thousandth (3 decimal places). P 35 6 R T Sin ∠ R = __________ = Cos ∠ R = __________ = Tan ∠ R = __________ = Sin ∠ P = __________ = Cos ∠ P = _________ = Tan ∠ P = _________ = III. For each of the trig ratios below fill in the correct angle: M 4 3 N 5 P 3 = sin ∠ _____ 5 Hall, Lepi 4 = cos ∠ _____ 5 DRAFT: 06/24/2003 4 = tan ∠ _____ 3 Farmington Public Schools 42 Integrated Math 2 Areas and Sectors name: Find the area of each of these circles: approximation.) (Write the area in terms of π and then use your calculator to find an 1. r = 5 cm. 2. r = 8 in. 3. diameter = 12 inches 4. R = 1.5 m. Complete this table. (Write the area in terms of π and then use your calculator to find an approximation.) Use correct units> Circle radius Circle diameter Area in terms of π Approximate area 11 feet 20 cm. 25 mm. 7 m. 32 ft. 1 mile Area of Sectors: Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 43 11. r = 8 cm. 12. r = 20 in. 80° 120° Area of sector = __________ Area of sector = __________ 13. d = 18 m. r = _________ Central Angle = 36° Area of sector = __________ Complete this table. Use correct units. Radius Diameter Central Angle 14 cm. 240° 5 feet 100° 30 mm. Area of sector 90° 60° 8 m. 12 in. 30° 20. What is the area of a quarter circle with a radius of 15 mm.? Hall, Lepi DRAFT: 06/24/2003 Farmington Public Schools 44