5.2 Trigonometric Ratios of Any Angle

Transcription

1 5. Tigonometic Ratios of An Angle The use of canes to lift heav objects is an essential pat of the constuction and shipping industies. Thee ae man diffeent designs of cane, but the usuall include some kind of winding mechanism, called a winch, which pulls o eleases a cable. In one cane design, the cable passes along a staight tube, called a boom o jib, which can be aised o loweed b changing its angle of inclination. The cable passes ove a pulle at the uppe end of the boom. When the hook at the end of the cable is attached to a heav object, the winch is used to aise o lowe it. Opeating a lage cane is a highl skilled occupation. In adjusting the position of the boom, the opeato uses the concepts of tigonomet, as will be shown in Eample 5. INVESTIGATE & INQUIRE Angles that measue 3, 45, and 6 occu often in tigonomet. The ae sometimes called special angles. Right tiangles in which the ae found ae sometimes called special tiangles.. ABC is an equilateal tiangle with side lengths of units. AD bisects BAC to fom two conguent tiangles, ABD and ACD. ABD and ACD ae called tiangles. What is the length of BD?. Detemine the length of AD. Epess the answe in adical fom. 3. Repeat questions and, beginning with equilateal tiangles with the following side lengths. Epess the length of AD as a mied adical. a) 4 units b) units 4. Descibe the elationship between the side lengths in an tiangle. B A 6 D C 5. Tigonometic Ratios of An Angle MHR 34

2 5. Use one of the tiangles to detemine the eact values of the sine, cosine, and tangent of 3 and DEF is an isosceles ight tiangle whose equal sides, DE and DF, have a length of unit. DEF is called a tiangle. Detemine the length of EF. Epess the answe in adical fom. 7. Repeat question 6, beginning with isosceles ight tiangles whose equal sides have the following lengths. Epess the length of EF as a mied adical. a) units b) 6 units 8. Descibe the elationship between the lengths of the sides in an tiangle. 9. Use one of the tiangles to detemine the eact values of the sine, cosine, and tangent of 45. F 45 E 45 D The angle is shown in standad position. The point (, ) is an point on the teminal am. B the thagoean theoem,. + Recall that ou have found the sine and cosine atios of an angle in standad position, whee 8. We will now etend the domain of to include angles of an measue, and will include the tangent atio of an angle in standad position. (,) Fo an angle in standad position, with point (, ) on the teminal am, and, + the thee pima tigonometic atios ae defined in tems of,, and as follows. sin cos tan EXAMLE Sine, Cosine, and Tangent of An Angle The point ( 3, 6) lies on the teminal am of an angle in standad position. Detemine the eact values of sin, cos, and tan. 34 MHR Chapte 5

3 SOLUTION Sketch the angle in standad position. Constuct AO b dawing a line segment pependicula to the -ais fom the point ( 3, 6). + + ( 3) + ( 6) sin cos tan o o A 3 O 6 ( 3, 6) So, sin, cos, and tan. 5 5 This chat summaizes the signs of tigonometic atios. Quadant II Quadant I 9 < < 8 < < 9 sin sin cos cos (,) tan + tan Quadant III Quadant IV 8 < < 7 7 < < 36 sin + sin + cos + cos tan + tan + 5. Tigonometic Ratios of An Angle MHR 343

4 The memo device CAST shows which tigonometic atios ae positive in each quadant. S sine T tangent A all C cosine The thagoean theoem shows that the side lengths of a tiangle have a atio of to to, o ::. The side lengths of a tiangle have a atio of to to, o ::. These special tiangles ae used to find the eact tigonometic atios of special angles. in Degees in Radians 6 π 4 π 3 π sin 3 cos 3 tan EXAMLE Eact Tigonometic Ratios fom Degee Measues Find the eact values of the sine, cosine, and tangent of. SOLUTION Sketch the angle in standad position. Constuct a tiangle b dawing a pependicula fom the teminal am to the -ais. The angle between the teminal am and the -ais is 6. So, the tiangle fomed is a tiangle. Choose point on the teminal am so that. It follows that and. 3 sin cos tan sin cos o tan o So, sin, cos, and tan. 344 MHR Chapte 5

5 EXAMLE 3 Eact Tigonometic Ratios fom Radian Measues Find the eact values of a) sin 7 4 π b) cos 5 6 π SOLUTION a) Sketch the angle in standad position. Constuct a tiangle b dawing a pependicula fom the teminal am to the -ais. 7 4 π ad 7 4 π 8 π 35 The angle between the teminal am and the -ais is 45. So, the tiangle fomed is a tiangle. Choose point on the teminal am so that. It follows that and. 7_ π 4 sin sin 7 4 π o So, sin 7 4 π. b) Sketch the angle in standad position. Constuct a tiangle b dawing a pependicula fom the teminal am to the -ais. 5 6 π ad 5 6 π 8 π 5 3 5_ π 6 The angle between the teminal am and the -ais is 3. So, the tiangle fomed is a tiangle. Choose point on the teminal am so that. It follows that and. cos cos 5 6 π o So, cos 5 6 π. 5. Tigonometic Ratios of An Angle MHR 345

6 EXAMLE 4 Tigonometic Ratios of a Right Angle Find the values of the sine, cosine, and tangent of an angle that measues 9. SOLUTION Sketch the angle in standad position. Choose (, ) on the teminal am of the angle. Theefoe,,, and. sin cos tan (,) sin 9 cos 9 tan 9 Since division b is not defined, tan 9 is not defined. 9 So, sin 9, cos 9, and tan 9 is not defined. EXAMLE 5 Opeating a Cane The boom of a cane can be moved so that its angle of inclination changes. In one location, close to some buildings and some ovehead powe cables, the minimum value of the angle of inclination is 3, and the maimum value is 6. If the boom is m long, find the vetical displacement of the end of the boom when the angle of inclination inceases fom its minimum value to its maimum value. Epess ou answe in eact fom and in appoimate fom, to the neaest tenth of a mete. m 6 3 vetical displacement SOLUTION a) In the diagam, the vetical displacement is. Using the eact values of the tigonometic atios fo 3 and 6, wite epessions fo and. sin 3 sin MHR Chapte 5

7 The vetical displacement is The vetical displacement is 5 5 m in eact fom, o 3.7 m in appoimate fom, to the neaest tenth of a mete. Ke Concepts Fo an angle in standad position, with point (, ) on the teminal am, and, + the thee pima tigonometic atios ae defined in tems of,, and as follows. sin cos tan The memo device CAST shows which tigonometic atios ae positive in each quadant. S sine T tangent A all C cosine The two special tiangles used to find the eact values of the sine, cosine, and tangent of special angles ae as shown tiangle tiangle Communicate You Undestanding. The point (3, 4) lies on the teminal am of an angle in standad position. Descibe how ou would detemine the eact values of sin, cos, and tan.. Descibe how ou would find the eact value of tan 3 π. 3. Descibe how ou would find the eact value of sin Descibe how ou would find the eact value of cos Tigonometic Ratios of An Angle MHR 347

8 actise A. The coodinates of a point on the teminal am of each in standad position ae shown, whee π. Detemine the eact values of sin, cos, and tan. a) b) (8,5) ( 3, 5) c) d) ( 4, 3) e) f) (, 7) (, 5) (3, ). The coodinates of a point on a teminal am of an in standad position ae given, whee π. Detemine the eact values of sin, cos, and tan. a) (6, 5) b) (, 8) c) (, 5) d) (6, ) e) (, 4) f) ( 3, 9) g) (3, 3) h) (, 6) 3. Find the eact value of each tigonometic atio. a) sin 3 b) tan 35 c) cos 4 d) tan 5 e) cos 5 f) sin 45 g) cos 33 h) sin 3 4. Find the eact value of each tigonometic atio. a) sin 5 4 π b) tan π 6 c) cos π 6 d) cos 7 4 π e) tan 4 3 π f) cos 7 6 π g) sin 5 6 π h) cos 3 4 π Appl, Solve, Communicate 5. Boom cane The am of a boom cane is m long. Because of the location of the constuction site, the angle of inclination of the boom of the cane has a minimum value of 3 and a maimum value of 45. Find the vetical displacement of the end of the boom as a) an eact value b) an appoimate value, to the neaest tenth of a mete 348 MHR Chapte 5

9 B 6. is in standad position with its teminal am in the stated quadant, and π. A tigonometic atio is given. Find the eact values of the othe two tigonometic atios. a) sin 4 5, quadant II b) cos 3, quadant III c) tan 5, quadant IV d) sin 3 7, quadant III 7. is in standad position, and π. A tigonometic atio is given. Find the eact values of the othe two tigonometic atios. a) sin 3 b) cos 3 5 c) tan 4 d) cos e) tan 8 5 f) sin Application A cago ship is tied up at a dock. At low tide, a -m long unloading amp slopes down fom the ship to the dock and makes an angle of 3 to the hoizontal. At high tide, the ship is close to the dock, and the unloading amp makes an angle of 45 to the hoizontal. a) Detemine the change in the hoizontal distance fom the ship to the dock fom low tide to high tide. Epess the distance as an eact value and as an appoimate value, to the neaest hundedth of a mete. b) Detemine the change in the height of the uppe end of the unloading amp above the dock fom low tide to high tide. Epess the change in height as an eact value and as an appoimate value, to the neaest hundedth of a mete. 9. Bascule bidges Bidges that ae hinged so that the can be aised ae known as bascule bidges. The ae commonl built ove wate, whee the can be opened to allow lage boats to pass though. Some bascule bidges ae hinged at one end, but othes ae built in two halves that sepaate when the ae aised. A bascule bidge built in two halves is 8 m long. When the bidge is opened full, each half of the bidge makes an angle of 6 with the hoizontal. When the bidge is opened full, how fa apat ae the top ends of the two halves of the bidge? m 8 m. Flagpole A flagpole is anchoed b two pais of gu wies attached 8 m above the gound. One pai is anchoed to the gound at 45 angles. The othe pai is anchoed to the gound at 6 angles. Which pai of gu wies has the geate total length and b how much, to the neaest tenth of a mete? 5. Tigonometic Ratios of An Angle MHR

10 . If A 36, find the possible measues of A. a) sin A b) cos A c) tan A d) sin A e) cos A f) tan A g) cos A h) sin A i) tan A j) tan A k) cos A. The table shows the tigonometic atios of a ight angle, fom Eample 4. Cop the table and complete it b finding the tigonometic atios of, 8, 7, and 36 angles. 3. If A π, and cos A, find the values of a) sin A b) tan A 4. If A π, and tan A, find the values of a) sin A b) cos A C 5. Inqui/oblem Solving If the shotest side of a tiangle is equal in length to the shotest side of a tiangle, what is the atio of the peimete of the fist tiangle to the peimete of the second tiangle, to the neaest hundedth? 6. If the longest side of a tiangle is equal in length to the shotest side of a tiangle, what is the atio of the aea of the fist tiangle to the aea of the second tiangle, to the neaest hundedth? 7. Communication The teminal am of angle in standad position has a slope of. If π, what ae the possible values of the following? Eplain. a) sin b) cos c) tan 8. Find the eact value of each of the following. a) sin 3 sin 45 sin 6 b) sin 3 cos 3 + sin 6 cos 6 c) sin 6 cos 3 + sin 3 cos 6 d) sin 3 cos 3 e) 5 tan 6 cos 3 Ratio sin cos tan 9 n.d. Angle Measue MHR Chapte 5