2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90


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1 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal am of angle detemining the value of sin, cos, o tan fo =, 9, 18, 7, o 36 solving fo all values of in an equation involving sine, cosine, and tangent solving a poblem involving tigonometic atios The Athabasca Oil Sands ae located 4 km noth of Fot McMua, AB. The ae the wold s lagest souce of snthetic cude fom oil sands, and the geatest single souce in Canada. Since the beginning of the fist oil sands poduction in 1967, technological advances have allowed fo a temendous incease in poduction and safet. Massive machine has been developed specificall fo the ecavation of the oil sands. Powe shovels ae equipped with a global positioning sstem (GPS) to make digging moe eact. The opeato must undestand the angles necessa to opeate the massive shovel. The design of powe shovels uses the laws of tigonomet. Did You Know? Man Canadian companies ae ve awae of and sensitive to concens about the impact of mining on the envionment. The companies consult with local Aboiginal people on issues such as the eestablishment of native tee species, like lowbush canbe and buffalo be. Investigate Tigonometic Ratios fo Angles Geate Than 9 Mateials gid pape potacto 1. On gid pape, daw a set of coodinate aes. a) Plot the point A(3, 4). In which quadant does the point A lie? b) Daw the angle in standad position with teminal am passing though point A. 88 MHR Chapte
2 . Daw a line pependicula to the ais though point A. 6 Label the intesection of this A(3, 4) line and the ais as point B. 4 This point is on the initial am of AOB. a) Use the Pthagoean O B 4 6 Theoem to detemine the  length of the hpotenuse,. 4 b) Wite the pima tigonometic atios fo. 6 c) Detemine the measue of, to the neaest degee. 3. How is each pima tigonometic atio elated to the coodinates of point A and the adius? 4. a) Reflect point A in the ais to obtain point C. Daw a line segment fom point C to the oigin. What ae the coodinates of point C? b) Daw a line pependicula to the ais though point C to ceate the efeence tiangle. Label the intesection of this line and the ais as point D. Use ou answes fom step 3 to wite the pima tigonometic atios fo COB. 5. a) What is the measue of COB, to the neaest degee? b) How ae COD and COB elated? Reflect and Respond 6. a) Compae the tigonometic atios fo AOB and COB. What ae the similaities and what ae the diffeences? b) Eplain wh some tigonometic atios ae positive and some ae negative. 7. a) Reflect point C in the ais to obtain point E. Which tigonometic atios would ou epect to be positive? Which ones would ou epect to be negative? Eplain ou easoning. b) Use the coodinates of point E and ou definitions fom step 3 to confim ou pediction. c) Etend this investigation into quadant IV. 8. Make a table showing the signs of the sine, cosine, and tangent atios of an angle,, in each of the fou quadants. Do ou notice a patten? How could ou ecognize the sign (positive o negative) of the tigonometic atios in the vaious quadants?. Tigonometic Ratios of An Angle MHR 89
3 Link the Ideas Finding the Tigonometic Ratios of An Angle, whee < 36 Suppose is an angle in standad position, and P(, ) is an point on its teminal am, at a distance fom the oigin. Then, b the Pthagoean Theoem, = +. You can use a efeence tiangle to detemine the thee pima tigonometic atios in tems of,, and. opposite sin = hpotenuse sin = adjacent cos = hpotenuse cos = tan = opposite adjacent tan = The chat below summaizes the signs of the tigonometic atios in each quadant. In each, the hoizontal and vetical lengths ae consideed as diected distances. P(, ) sin = Quadant II 9 < < 18 cos =  tan = sin > cos < tan <  sin = Quadant I < < 9 cos = tan = sin > cos > tan > P(, ) P(, ) R Wh is alwas positive?  R = 18  R = R Quadant III 18 < < 7 sin =  cos =  tan =   sin < cos < tan > sin =  Quadant IV 7 < < 36 cos = tan =  sin < cos > tan <   R R  P(, ) P(, ) = 18 + R = 36  R 9 MHR Chapte
4 Eample 1 Wite Tigonometic Ratios fo Angles in An Quadant The point P(8, 15) lies on the teminal am of an angle,, in standad position. Detemine the eact tigonometic atios fo sin, cos, and tan. Solution Sketch the efeence tiangle b dawing a line pependicula to the ais though the point (8, 15). The point P(8, 15) is in quadant II, so the teminal am is in quadant II. Use the Pthagoean Theoem to detemine the distance,, fom P(8, 15) to the oigin, (, ). = + = (8) + (15) = 89 = 17 P(8, 15) The tigonometic atios fo can be witten as follows: sin = cos = tan = sin = 15 cos = 8 tan = cos =  8 tan = You Tun The point P(5, 1) lies on the teminal am of an angle,, in standad position. Detemine the eact tigonometic atios fo sin, cos, and tan R Eample Detemine the Eact Value of a Tigonometic Ratio Detemine the eact value of cos 135. Solution The teminal am of 135 lies in quadant II. The efeence angle is , o 45. The cosine atio is negative in quadant II. cos 135 =  1 Wh ae side lengths 1, 1, and used? You Tun Detemine the eact value of sin 4.. Tigonometic Ratios of An Angle MHR 91
5 Eample 3 Detemine Tigonometic Ratios Suppose is an angle in standad position with teminal am in quadant III, and cos =  3. What ae the eact values of sin 4 and tan? Solution Sketch a diagam R Use the definition of cosine to find the eact values of and. cos = cos = Since the teminal am is in quadant III, is negative. is alwas positive. So, = 3 and = 4. Use = 3, = 4 and the Pthagoean Theoem to find. + = (3) + = = 16 = 169 = 7 = ± 7 = 7 is a solution fo = 7 because ( 7 )( 7 ) = 7 =  Use = 3, =  7, and = 4 to wite sin and tan. sin = tan = sin =  7 tan = sin =  7 tan = You Tun 7 is also a solution because (  7 )(  7 ) = 7 Suppose is an angle in standad position with teminal am in quadant III, and tan = 1. Detemine the eact values of sin and cos. 5 Wh is  7 used fo hee? 9 MHR Chapte
6 Eample 4 Detemine Tigonometic Ratios of Quadantal Angles Detemine the values of sin, cos, and tan when the teminal am of quadantal angle coincides with the positive ais, = 9. Solution Let P(, ) be an point on the positive ais. Then, = and =. = P(, ) = 9 quadantal angle an angle in standad position whose teminal am lies on one of the aes eamples ae, 9, 18, 7, and 36 The tigonometic atios can be witten as follows. sin 9 = cos 9 = tan 9 = sin 9 = cos 9 = tan 9 = sin 9 = 1 cos 9 = tan 9 is undefined You Tun Use the diagam to detemine the values of sin, cos, and tan fo quadantal angles of, 18, and 7. Oganize ou answes in a table as shown below. 9 (, ), = Wh is tan 9 undefined? (, ), = 18 (, ), = 7 (, ), = sin 1 cos tan undefined. Tigonometic Ratios of An Angle MHR 93
7 Solving fo Angles Given Thei Sine, Cosine, o Tangent Step 1 Detemine which quadants the solution(s) will be in b looking at the sign (+ o ) of the given atio. Wh ae the tigonometic atios fo Step Solve fo the efeence angle. the efeence angle alwas positive? Step 3 Sketch the efeence angle in the appopiate quadant. Use the diagam to detemine the measue of the elated angle in standad position. Eample 5 Solve fo an Angle Given Its Eact Sine, Cosine, o Tangent Value Solve fo. a) sin =.5, < 36 b) cos =  3, < 18 Solution a) Since the atio fo sin is positive, the teminal am lies in eithe quadant I o quadant II. sin R =.5 How do ou know R = 3 R = 3? In quadant I, = 3. In quadant II, = 183 = 15 The solution to the equation sin =.5, < 36, is = 3 o = 15. II 3 III 3 I IV b) Since the cosine atio is negative, the teminal am must lie in quadant II o quadant III. Given the estiction < 18, the teminal am must lie in quadant II. Use a tiangle to detemine the efeence angle, R. cos R = 3 R = Using the efeence angle of 3 in quadant II, the measue of is 183 = 15. The solution to the equation cos =  3, < 18, is = You Tun Solve sin =  1, < MHR Chapte
8 Eample 6 Solve fo an Angle Given Its Appoimate Sine, Cosine, o Tangent Value Given cos = , whee < 36, detemine the measue of, to the neaest tenth of a degee. Solution The cosine atio is negative, so the angles in standad position lie in quadant II and quadant III. Use a calculato to detemine the angle that has cos R = R = cos 1 (.6753) Wh is cos 1 (.6753) the efeence angle? R 47.5 With a efeence angle of 47.5, the measues of ae as follows: In quadant II: In quadant III: = = = 13.5 = 7.5 You Tun Detemine the measue of, to the neaest degee, given sin = .89, whee < 36. Ke Ideas The pima tigonometic atios fo an angle,, in standad position that has a point P(, ) on its teminal am ae sin =, cos =, and tan =, whee = +. The table show the signs of the pima tigonometic atios fo an angle,, in standad position with the teminal am in the given quadant. Quadant Ratio I II III IV sin cos tan If the teminal am of an angle,, in standad position lies on one of the aes, is called a quadantal angle. The quadantal angles ae, 9, 18, 7, and 36, 36.. Tigonometic Ratios of An Angle MHR 95
9 Check You Undestanding Pactise 1. Sketch an angle in standad position so that the teminal am passes though each point. a) (, 6) b) (4, ) c) (5, ) d) (1, ). Detemine the eact values of the sine, cosine, and tangent atios fo each angle. a) c) 15 6 b) 5 d) 9 3. The coodinates of a point P on the teminal am of each angle ae shown. Wite the eact tigonometic atios sin, cos, and tan fo each. a) c) P(3, 4) P(8, 15) b) P(1, 5) d) P(1, 1) 4. Fo each desciption, in which quadant does the teminal am of angle lie? a) cos < and sin > b) cos > and tan > c) sin < and cos < d) tan < and cos > 5. Detemine the eact values of sin, cos, and tan if the teminal am of an angle in standad position passes though the given point. a) P(5, 1) b) P(5, 3) c) P(6, 3) d) P(4, 1) 6. Without using a calculato, state whethe each atio is positive o negative. a) sin 155 b) cos 3 c) tan 1 d) cos 7. An angle is in standad position such that sin = a) Sketch a diagam to show the two possible positions of the angle. b) Detemine the possible values of, to the neaest degee, if < An angle in standad position has its teminal am in the stated quadant. Detemine the eact values fo the othe two pima tigonometic atios fo each. Ratio Value Quadant a) cos =  3 II b) sin = 3 5 I c) tan = IV d) sin = III e) tan = 1 III 96 MHR Chapte
10 9. Solve each equation, fo < 36, using a diagam involving a special ight tiangle. a) cos = 1 c) tan =  1 b) cos =  1 d) sin = e) tan = 3 f) tan = Cop and complete the table using the coodinates of a point on the teminal am. sin cos tan Detemine the values of,,, sin, cos, and tan in each. a) P(8, 6) b) R R P(5, 1) Appl 1. Point P(9, 4) is on the teminal am of an angle. a) Sketch the angle in standad position. b) What is the measue of the efeence angle, to the neaest degee? c) What is the measue of, to the neaest degee? 13. Point P(7, 4) is on the teminal am of an angle,. a) Sketch the angle in standad position. b) What is the measue of the efeence angle, to the neaest degee? c) What is the measue of, to the neaest degee? 14. a) Detemine sin when the teminal am of an angle in standad position passes though the point P(, 4). b) Etend the teminal am to include the point Q(4, 8). Detemine sin fo the angle in standad position whose teminal am passes though point Q. c) Etend the teminal am to include the point R(8, 16). Detemine sin fo the angle in standad position whose teminal am passes though point R. d) Eplain ou esults fom pats a), b), and c). What do ou notice? Wh does this happen? 15. The point P(k, 4) is 5 units fom the oigin. If P lies on the teminal am of an angle,, in standad position, < 36, detemine a) the measue(s) of b) the sine, cosine, and tangent atios fo 16. If cos = 1 5 and tan = 6, detemine the eact value of sin. 17. The angle between the hoizontal and Eath s magnetic field is called the angle of dip. Some migato bids ma be capable of detecting changes in the angle of dip, which helps them navigate. The angle of dip at the magnetic equato is, while the angle at the Noth and South Poles is 9. Detemine the eact values of sin, cos, and tan fo the angles of dip at the magnetic equato and the Noth and South Poles.. Tigonometic Ratios of An Angle MHR 97
11 18. Without using technolog, detemine whethe each statement is tue o false. Justif ou answe. a) sin 151 = sin 9 b) cos 135 = sin 5 c) tan 135 = tan 5 d) sin 6 = cos 33 e) sin 7 = cos Cop and complete the table. Use eact values. Etend the table to include the pima tigonometic atios fo all angles in standad position, 9 36, that have the same efeence angle as those listed fo quadant I. sin cos tan Albeta Aboiginal Touism designed a cicula icon that epesents both the Métis and Fist Nations communities of Albeta. The cente of the icon epesents the collection of all peoples pespectives and points of view elating to Aboiginal histo, touching eve quadant and diection. a) Suppose the icon is placed on a coodinate plane with a efeence angle of 45 fo points A, B, C, and D. Detemine the measue of the angles in standad position fo points A, B, C, and D. b) If the adius of the cicle is 1 unit, detemine the coodinates of points A, B, C, and D. B A 1. Eploe pattens in the sine, cosine, and tangent atios. a) Cop and complete the table stated below. List the sine, cosine, and tangent atios fo in incements of 15 fo 18. Whee necessa, ound values to fou decimal places. Angle Sine Cosine Tangent b) What do ou obseve about the sine, cosine, and tangent atios as inceases? c) What compaisons can ou make between the sine and cosine atios? d) Detemine the signs of the atios as ou move fom quadant I to quadant II. e) Descibe what ou epect will happen if ou epand the table to include quadant III and quadant IV. Etend. a) The line = 6, fo, ceates an acute angle,, with the ais. Detemine the sine, cosine, and tangent atios fo. b) If the teminal am of an angle,, lies on the line =, fo, detemine the eact value of tan + cos. 3. Conside an angle in standad position with = 1 cm. Descibe how the measues of,, sin, cos, and tan change as inceases continuousl fom to 9. 1 cm C D 98 MHR Chapte
12 4. Suppose is a positive acute angle and cos = a. Wite an epession fo tan in tems of a. 5. Conside an angle of 6 in standad position in a cicle of adius 1 unit. Points A, B, and C lie on the cicumfeence, as shown. Show that the lengths of the sides of ABC satisf the Pthagoean Theoem and that CAB = A 6 C B Ceate Connections 6. Eplain how ou can use efeence angles to detemine the tigonometic atios of an angle,. 7. Point P(5, 9) is on the teminal am of an angle,, in standad position. Eplain the ole of the efeence tiangle and the efeence angle in detemining the value of. 8. Eplain wh thee ae eactl two nonquadantal angles between and 36 that have the same sine atio. 9. Suppose that is an angle in standad position with cos =  1 and sin =  3, < 36. Detemine the measue of. Eplain ou easoning, including diagams. 3. MINI LAB Use dnamic geomet softwae to eploe the tigonometic atios. Step 1 a) Daw a cicle with a adius of 5 units and cente at the oigin. b) Plot a point A on the cicle in quadant I. Join point A and the oigin b constucting a line segment. Label this distance. Step a) Recod the coodinate and the coodinate fo point A. b) Constuct a fomula to calculate the sine atio of the angle in standad position whose teminal am passes though point A. Use the measue and calculate featues of ou softwae to detemine the sine atio of this angle. c) Repeat step b) to detemine the cosine atio and tangent atio of the angle in standad position whose teminal am passes though point A. Step 3 Animate point A. Use the motion contolle to slow the animation. Pause the animation to obseve the atios at points along the cicle. Step 4 a) What obsevations can ou make about the sine, cosine, and tangent atios as point A moves aound the cicle? b) Recod whee the sine and cosine atios ae equal. What is the measue of the angle at these points? c) What do ou notice about the signs of the atios as point A moves aound the cicle? Eplain. d) Fo seveal choices fo point A, divide the sine atio b the cosine atio. What do ou notice about this calculation? Is it tue fo all angles as A moves aound the cicle?. Tigonometic Ratios of An Angle MHR 99
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