GZW. How can you find exact trigonometric ratios?

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1 4. Special Angles Aircraft pilots often cannot see other nearb planes because of clouds, fog, or visual obstructions. Air Traffic Control uses software to track the location of aircraft to ensure that the are kept a safe distance from one another. The software uses trigonometr to make these calculations. The radar screen here shows an aircraft, identified b GZW, 0 km east of the control tower, and another, identified as TGL, 8 km southwest of the tower. To find the distance between the two aircraft, the software can use the cosine law, but it needs the cosine of an obtuse angle. Are there trigonometric ratios for angles greater than 90? If so, how are the calculated? In this section, ou will learn how to find the primar trigonometric ratios for an angle from 0 to 360. TGL 8 km 0 km GZW Tools computer with The Geometer s Sketchpad or grid paper unit circle a circle with centre at the origin and a radius of unit Technolog Tip In The Geometer s Sketchpad, ou can drag the unit point on the -ais to make the circle a convenient size. When measuring distances using the Measure menu, be sure to select Coordinate Distance to appl the proper scale factor. Investigate A How can ou find eact trigonometric ratios? Some triangles contain known angles and sides. You can use these triangles to find eact trigonometric ratios for special angles.. Draw a set of aes. Using the origin as the centre, draw a circle. The radius of this circle will represent unit. This circle is known as a unit circle. If ou are using grid paper for this investigation, create a unit circle of workable size to help with accurac. Second 0 B Third First A Fourth 222 MHR Functions Chapter 4

2 2. a) Draw a 45 angle in the first quadrant b placing the initial arm on the -ais. Etend the terminal arm of the angle until it meets the circle at point A, such that OA is a radius of the circle. This representation is known as an angle in standard position. b) Draw a vertical line from point A to the -ais, and label the intersection point B. c) Draw a line from the origin to point B to form noab. 3. Reflect Classif noab. Be as specific as ou can. 4. Use the Pthagorean theorem to find the side lengths of noab. Leave our answers in radical form. Do not convert to a decimal. 5. a) Use the side lengths to find eact epressions for sin 45, cos 45, and tan 45. b) Use a calculator to evaluate these epressions to 4 decimal places. 6. Use a calculator to determine the values of sin 45, cos 45, and tan a) Reflect How do the trigonometric ratios obtained from the triangle compare to those obtained from the calculator? b) What is the relationship between cos 45 and the measure of side OB of noab? c) What is the relationship between sin 45 and the measure of side AB of noab? d) Find the coordinates of point A. How do the two relationships in parts b) and c) relate to the coordinates of point A? Investigate B How can ou find trigonometric ratios for angles greater than 90?. Line segment OA from Investigate A forms an angle of 45 with the -ais. Reflect point A in the -ais to obtain point C, and join point C to the origin. What is the measure of the angle between OC and the negative -ais? C A This angle is referred to as a 45 reference angle. The significance of knowing the reference angle 0 B is that the values of the si trigonometric functions for an angle greater than 90 are the same as the corresponding values for its reference angle with a possible change in sign. initial arm first arm, or ra, of an angle drawn on a Cartesian plane that meets the other (terminal) arm of the angle at origin terminal arm the arm of an angle that meets the initial arm at the origin and rotates around the origin counterclockwise to form a positive angle or clockwise to form a negative angle 0 Terminal Arm Initial Arm angle in standard position the position of an angle when its initial arm is on the positive -ais and its verte is at the origin Technolog Tip In The Geometer s Sketchpad, ou can use the Transform menu to rotate the terminal arm about the origin to form an eact angle. For help with The Geometer s Sketchpad, refer to the Technolog Appendi on pages 496 to 56. reference angle the acute angle between the terminal arm and the -ais of an angle in standard position 4. Special Angles MHR 223

3 2. What are the coordinates of point C? 3. Look back at the relationships found in step 7d) of Investigate A. Use the coordinates of point C to determine cos 35 and sin Reflect How can ou use the coordinates of point C to represent tan 35? 5. a) Find tan 35. b) Find the slope of the terminal arm OC. How does this slope relate to tan 35? Connections How does a calculator find trigonometric ratios for angles? The sine of an angle can be epressed as a series of values involving powers of and various coefficients. The calculator adds the terms of the series to find the sine, using enough terms to obtain the desired accurac. Similar series eist for cosine and tangent. These are known as Talor series. Visit the McGraw-Hill Rerson Web site and follow the links to learn more about Talor series. 6. Use a calculator to compare the trigonometric ratios that ou found in steps 3 and 5 with the calculator values of sin 35, cos 35, and tan 35. Investigate C How can ou use a unit circle to find the trigonometric ratios for an angle?. Starting from the -ais, plot a point D such that the line OD forms an angle of 230 measured counterclockwise from the positive -ais. 2. To find the coordinates of point D, draw a vertical line to meet the -ais at point E. The angle EOD is the reference angle for the angle in standard position. 3. Measure OE and ED. Use the measurements to determine the coordinates of point D. Record the coordinates of point D. E Find sin 230, cos 230, and tan 230. D 5. Reflect Eplain wh sin 230 and cos 230 are negative, but tan 230 is positive. 224 MHR Functions Chapter 4

4 Eample Determine the Primar Trigonometric Ratios for 30 and 60 Another triangle whose side lengths and angles are known is an equilateral triangle. a) Draw an equilateral triangle with side length 2 units such that the base is horizontal. From the top verte, draw a vertical line to form two congruent right triangles. b) What are the measures of the angles in these triangles? c) Find the side lengths of the base and height of one of these triangles. Leave answers in radical form where appropriate. d) Use the side lengths and angle measures to find eact values of the trigonometric ratios for 30 and 60. Solution a) b) The altitude of the triangle bisects the top angle into two 30 angles. The angles in each triangle are 30, 60, and 90. c) Let represent the base of one of the triangles. is half the length of one side, or unit. Let represent the height of the triangle. Since the triangle is a right triangle, the Pthagorean theorem applies Since lengths are positive, discard the negative value for. d) Since the adjacent side to the 60 angle measures unit, the opposite side measures 3 units, and the hpotenuse measures 2 units, sin 60 5 _ 3, cos 60 5 _ 2 2, and tan 60 5 _ Similarl, sin 30 5 _, cos 30 5 _ 3 2 2, and tan 30 5 _ 3 Technolog Tip A computer algebra sstem (CAS) can displa either eact or approimate values of the trigonometric ratios for special angles. You will learn more about how to use a CAS in the Use Technolog section following Section Special Angles MHR 225

5 Eample 2 Trigonometric Ratios for 0, 90, 80, and 270 Use a unit circle to find eact values of the trigonometric ratios for 0, 90, 80, and 270. Connections Solution θ is the lowercase form of the Greek letter theta. Greek letters are often used to represent variable quantities in science and mathematics. Other letters often used for angles are α, β, and φ (alpha, beta, and phi). Choose a point on the terminal arm of each angle in the unit circle. For an angle of 0, the required point is on the -ais at (, 0). sin θ 5 cos θ 5 tan θ 5 _ sin cos 0 5 tan 0 5 0_ For an angle of 90, the required point is on the -ais at (0, ). sin θ 5 cos θ 5 tan θ 5 _ sin 90 5 cos tan 90 5 _ 0 tan 90 is undefined. Division b 0 is undefined. For an angle of 80, the required point is on the -ais at (, 0). sin θ 5 cos θ 5 tan θ 5 _ sin cos 80 5 tan _ tan For an angle of 270, the required point is on the -ais at (0, ). sin θ 5 cos θ 5 tan θ 5 _ sin cos tan _ 0 tan 270 is undefined. Connections To represent 0 west of north, start from north and turn 0 toward the west. West 0 North Eample 3 Appl Trigonometric Ratios An air traffic controller observes that a ValuAir flight is 20 km due east of the control tower, while a First Class Air flight is 25 km in a direction 0 west of north from the control tower. a) What is the angle of separation of the two aircraft as seen from the tower? b) Construct a unit circle to determine the cosine of the angle in part a). 226 MHR Functions Chapter 4

6 Solution a) From east to north is an angle of 90. A further angle of 0 results in an angle of separation of 00. b) Use geometr software or grid paper to construct a unit circle, and plot the point A required for an angle of 00. Draw a vertical line to meet the -ais at point B to complete the triangle. Measure the sides of the triangle to determine the coordinates of point A. A Te c h n o l o g You can obtain accurac to about one decimal place when using grid paper. Using geometr software generall allows more accurac. With The Geometer s Sketchpad, for eample, ou can set the Preferences under the Edit menu to measure up to five decimal places B Reminder: If ou are using grid paper, appl an appropriate scale factor. Tip The coordinates of point A are approimatel ( 0.7, 0.98). Therefore, cos Ke Concepts Using a unit circle is one wa to find the trigonometric ratios for angles greater than 90. An point on a unit circle can be joined to the origin to form the terminal arm of an angle. The angle θ is measured starting from the initial arm along the positive -ais, proceeding counterclockwise to the terminal arm. The coordinates of the point (, ) on a unit circle are related to θ such that 5 cos θ and 5 sin θ. tan θ 5 _ Eact trigonometric ratios for special angles can be determined using special triangles (, ) θ 0 45, cos 45 5 _, and tan The eact trigonometric ratios for 45 are sin 45 5 _ Special Angles MHR 227 Functions CH04.indd 227 6/0/09 4:08: PM

7 Communicate Your Understanding C As the terminal arm moves counterclockwise from the positive -ais around the circle, trace what happens to the sign of cos θ as ou move from 0 to 360. Eplain wh this happens in terms of coordinates. Then, do the same trace for sin θ. Finall, trace what happens for tan θ. C2 Some trigonometric ratios for certain angles are undefined. Give two eamples. Eplain wh the are undefined. C3 Which trigonometric ratios are positive in the fourth quadrant? Which are negative? Eplain wh. Second First 0 Third Fourth A Practise For help with questions to 4, see Investigate A and Eamples and 2.. Compare the eact values of the trigonometric ratios for 30 and 60 to the trigonometric ratios calculated b a calculator. 2. Compare the eact values of the trigonometric ratios from Eample 2 to the trigonometric ratios calculated b a calculator. 3. a) Use a unit circle to represent an angle of 30. Draw a triangle and use it to write the three primar trigonometric ratios in eact form for 30. b) Use a unit circle to represent an angle of 60. Draw a triangle and use it to determine the eact primar trigonometric ratios for In a table, summarize the eact trigonometric ratios for the angles 0, 30, 45, 60, and 90. Add and complete a column for the ratios as given b a calculator, correct to 4 decimal places. For help with questions 5 and 6, see Investigate B. 5. a) When using a unit circle to find trigonometric ratios for 35, a reference angle of 45 is used. What reference angle should ou use to find the trigonometric ratios for 20? b) Construct a unit circle to find the eact values of the three primar trigonometric ratios for Construct a unit circle to find the eact values of the three primar trigonometric ratios for 35. For help with questions 7 and 8, see Investigate C and Eample Use a unit circle to find the approimate primar trigonometric ratios for 40. Measure an side lengths needed. Compare our answers to those generated b calculator, correct to 4 decimal places. 8. Use a unit circle to find the approimate primar trigonometric ratios for 30. Measure an side lengths needed. Compare our answers to those generated b calculator. 9. Create a table to summarize the eact values of the primar trigonometric ratios for 0, 90, 80, 270, and MHR Functions Chapter 4

8 0. a) Which trigonometric ratios are positive for angles in the first quadrant? second quadrant? third quadrant? fourth quadrant? B b) One wa to remember the signs of trigonometric S A ratios is called T 0 C the CAST rule, as shown (the letters spell CAST, moving counterclockwise, beginning in the fourth quadrant. What do the letters in each quadrant stand for? Connect and Appl. A pine tree that is 0 m tall is damaged in a windstorm such that it leans sidewas to make an angle of 60 with the ground. a) Represent this situation with a diagram. b) Find an eact epression for the length of the shadow of the tree when the sun is directl overhead. 2. A sailboat is 2 km north of a lighthouse. A motor cruiser is 2 km east of the same lighthouse. a) Use trigonometr to find an eact epression for the distance between the two boats. b) Check our answer using another method. 3. Tall structures Reasoning and Proving are sometimes stabilized with Representing Selecting Tools ropes or cables Problem Solving attached to the Connecting Reflecting ground. These Communicating stabilizers are known as gu wires. A flagpole is stabilized b two gu wires attached to the top of the pole. On one side, a 25-m-long wire makes an angle of 60 with the ground. The sine of the angle formed b the second wire and the ground equals the cosine of the angle of the first gu wire. a) Represent this situation with a diagram. b) Determine the length, to the nearest tenth of a metre, of the second gu wire without calculating an angles. c) Wh is it not necessar to find the angle that the second gu wire makes with the ground to solve the problem? d) Determine the angle made b the second gu wire with the ground. 4. Use Technolog Use a calculator for this question. a) Cop and complete the table. θ sin θ Sign b) Relate the sign of sin θ with the quadrant. Are the signs as ou epected? c) Now, work backward. Find the angle that satisfies i) sin θ ii) sin θ The calculator will give ou onl one answer for each, despite the fact that there are two angles between 0 and 360 that have each value. Note also that the calculator epressed the second angle as 30 and not 330. For angles between 80 and 360, the calculator starts at the positive -ais and proceeds in a clockwise direction. Angles measured in this direction are defined as negative. d) Construct a similar table for cos θ, using the angles 60, 20, 240, and 300. Then, start with the cosine, and find the angle for both positive and negative values. Note the answers provided b the calculator. e) Select suitable angles to test tan θ. Note how the calculator presents the angles when ou work backward. 4. Special Angles MHR 229

9 5. a) Pose a real-world problem that can be solved using trigonometric Reasoning and Proving Representing Problem Solving ratios for special Connecting Reflecting angles without Communicating using a calculator. Solve our problem to ensure that there is a solution. Selecting Tools b) Trade our problem with a classmate. Solve each other s problem. c) Trade solutions. Judge the mathematical correctness of the solution. Look for proper form and careful use of mathematical smbols. 6. Chapter Problem You are about to begin Chantal s trigonometric orienteering course. Prepare a set of aes on grid paper to make a map of our progress, labelling the Start position at the origin. All coordinates for the course will be positive. You ma not use a calculator to help ou until ou reach checkpoint #3. Use the instructions below to calculate the direction and distance to checkpoint #. Draw this leg on our 4 map and label the 3 angle and distance m Choose and record a 6 suitable scale. Start Checkpoint # 20 m Direction: North of east Use the angle in the first quadrant that has a sine of _ 2. Distance: The result of evaluating 40(cos 50 tan 35 sin 300 ) 7. Stefan has set up a right nefg on one side of a river such that FG measures 20 m and /DEF measures D 60. EG bisects /DEF. Without using a calculator, determine the width, DG, of the river. E 60 G 20 m F Achievement Check 8. In npqr, /Q 5 90, /P 5 60, and /R PR 5 unit. Etend side QR to T such that PR 5 RT. Join PT. a) Draw a diagram to represent this situation. b) Calculate the eact measure of /T. Justif our answer. c) What lengths do ou need to know to find tan T? Eplain. d) Determine the eact value of the unknown lengths in part c). Do not use a calculator. Justif our reasoning. e) Find the eact value for tan T. C Etend 9. Consider an angle of 30 in standard position on a unit circle. Join A to B and to C as shown. Show that the lengths of the sides of nabc satisf the Pthagorean theorem and that /CAB C 30 0 B 20. Refer to question 9. Let /AOB be an angle in the first quadrant, and let the coordinates of A be (, ). Show that the sides of nabc satisf the Pthagorean theorem and that /CAB A 230 MHR Functions Chapter 4

10 2. The town of Dainfleet is planning to build a municipal swimming pool in the shape of a regular heagon. The projected cost of the pool depends on its area. Without using a calculator, show that the side length,, of the pool is related to its area, A, b the formula 2A 5 _ Math Contest An equilateral triangle has a height of 3 3 cm. Its perimeter is A 2 cm C 6 cm B 8 cm D 9 3 cm E cm 23. Math Contest The parallel sides of a trapezoid have lengths of 7 cm and 5 cm. The two lower base angles are 30 and 60. The area of the trapezoid is A 22 3 cm2 B 4 3 cm2 D 30 3 cm2 2 C 22 cm E 8 3 cm2 24. Math Contest A circle has an inscribed isosceles triangle with one side as the diameter. What is the ratio of the area of the triangle to the area of the circle? π:2 C :π A B :2π D 2:π E :4 Career Connection Building codes eist to ensure that structures are properl built with suitable building materials. As a building inspector, Christopher visits sites during all phases of the construction to make sure regulations are being followed. Initiall, a site must be able to support the tpe of building planned and blueprints have to be checked for the structure s stabilit. As the building goes up, Christopher checks the stabilit, wiring, and safet features. A good knowledge of trigonometr is important when buildings are meant to last and when people s safet is at stake. To prepare for his job academicall, Christopher completed a three-ear diploma in architecture and construction engineering technolog at Conestoga College. 4. Special Angles MHR 23 Functions CH04.indd 23 6/0/09 4:08:20 PM