Lesson 6.2 Exercises, pages

Transcription

1 Lesson 6.2 Eercises, pages A. Sketch each angle in standard position. a) 7 b) 40 Since the angle is between Since the angle is between 0 and 90, the terminal 90 and 80, the terminal arm is in Quadrant. arm is in Quadrant c) 200 d) 40 Since the angle is between Since the angle is between 80 and 270, the terminal 270 and 60, the terminal arm is in Quadrant. arm is in Quadrant P D NT CPY. 9

2 4. a) Determine the reference angle for each angle in standard position. i) 4 ii) 98 The angle is in Quadrant, The angle is in Quadrant 2, so its reference angle is 4. so its reference angle is: iii) 24 iv) 290 The angle is in Quadrant, The angle is in Quadrant 4, so its reference angle is: so its reference angle is: b) For each angle in part a, determine the other angles between 90 and 60 that have the same reference angle. i) 4 ii) 98 The angles with the same The angles with the same reference angle are: reference angle are: iii) 24 iv) 290 The angles with the same The angles with the same reference angle are: reference angle are: Determine the quadrant in which the terminal arm of each angle in standard position lies. a) 280 b) 88 Since the angle is between Since the angle is between 270 and 60, the terminal 0 and 90, the terminal arm lies in Quadrant 4. arm lies in Quadrant. c) 9 d) 0 Since the angle is between Since the angle is between 80 and 270, the terminal 90 and 80, the terminal arm lies in Quadrant. arm lies in Quadrant 2. 0 D NT CPY. P

3 B 6. Point P(, 4) is a terminal point of an angle u in standard position. a) Sketch the angle. P(, 4) Plot P(, 4) ; draw a line through P. Label U. Let the length of P r. r b) What is the distance from the origin to P? Use: r 2 2 Substitute:, 4 r ( ) r 2 r c) Write the primar trigonometric ratios of u., 4, r sin U r 4 cos U r tan U, or 4, or 4 d) To the nearest degree, what is u? Use: sin U 4 sin a 4 b U is approimatel 80, or 7. Each angle u is in standard position. State the quadrants in which the terminal arm of the angle could lie. a) cos u = - 2 b) tan u = -4 cos U, so is negative; tan U r, so if is negative, and the terminal arm lies the terminal arm lies in in Quadrant 2 or. Quadrant 2; if is negative, the terminal arm lies in Quadrant 4. c) sin u = 2 d) tan u = 0.2 sin U, so is positive; tan U r, so if both and and the terminal arm lies are positive, the terminal arm in Quadrant or 2. lies in Quadrant ; if both and are negative, the terminal arm lies in Quadrant. P D NT CPY.

4 8. a) Choose an angle between 90 and 80. Sketch a diagram to show that the angle can be formed b reflecting its reference angle in an ais or aes. Sample response: For, its reference angle is: 80 6 Draw 6 in Quadrant. Choose a terminal point, P. Reflect P in the -ais. Then P is the terminal arm of. P P 6 b) Repeat part a for an angle between 80 and 270. Sample response: For 220, its reference angle is: Draw 40 in Quadrant. P Choose a terminal point, P. Reflect P in the -ais, then reflect P in the -ais. 220 Then P is the terminal arm of 220. P 40 P c) Repeat part a for an angle between 270 and 60. Sample response: For 0, its reference angle is: Draw 0 in Quadrant. Choose a terminal point, P. Reflect P in the -ais. Then P is the terminal arm 0 of 0. 0 P P 9. Determine possible coordinates of a terminal point for each angle in standard position. a) Sample response: Choose a value for r, such as r 4. Then: r cos r sin 4 4a b, or 4a b, or Possible coordinates are: a 4, 4 b D NT CPY. P

5 b) 20 Sample response: Choose a value for r, such as r. Then: r cos 20 r sin 20, or a a, or 2 b 2 2 b 2 Possible coordinates are: a 2, 2 b c) 20 Sample response: Choose a value for r, such as r 6. Then: r cos 20 r sin 20 6a, or 6a, or 2 b Possible coordinates are: (, 2 b ) 0. Each point below lies on the terminal arm of an angle u in standard position. Determine: i) cos u ii) sin u iii) tan u iv) the value of u to the nearest degree a) A(, 4) b) B( 6, 0) r 2 2 r ( ) 2 ( 4) 2 r r 2 2 r ( 6) 2 (0) 2 r 6 i) cos U r ii) sin U r i) cos U r ii) sin U r iii) tan U iii) tan U 4 4, or 0, or 0 6 iv) Use: tan U 4 iv) Since the terminal point lies on the negative -ais, U 80 tan a 4 b.0... Since both and are negative, the terminal arm lies in Quadrant, and U is approimatel: 80 2 P D NT CPY.

6 c) C(0, 2) d) D(2, ) r 2 2 r (0) 2 (2) 2 r 2 r 2 2 r (2) 2 ( ) 2 r i) cos U r ii) sin U r i) cos U r ii) sin U r iii) tan U 2 0 tan U is undefined. iv) Since the terminal point is on the positive -ais, U 90. A hiker sketches a map of her destination, D, from her starting point,. The hiker can travel onl west, then north. To the nearest kilometre, how far must she hike to get to her destination? iii) tan U, or 2 2 iv) Use: cos U 2 cos a 2 b Since is positive and is negative, the terminal arm lies in Quadrant 4 and U is approimatel: W D 0 km N S 40 E The distance the hiker travels west is the absolute value of the -coordinate of D. r cos U Substitute: r 0 and U 40 0 cos The distance the hiker travels north is the -coordinate of D. r sin U Substitute: r 0 and U 40 0 sin In kilometres, the hiker travels: The hiker travels approimatel 4 km. 4 D NT CPY. P

7 2. Eplain wh (sin u) 2 + (cos u) 2 = for an angle u in standard position in Quadrant 2. In Quadrant 2, cos U and sin U r r So, (sin U) 2 (cos U) 2 a 2 r b a 2 r b Since 2 2 r 2, r r r 2 (sin U) 2 (cos U) 2 r2 r 2. The location of the terminal arm of an angle a and a trigonometric ratio of its reference angle u are given. a) The terminal arm lies in Quadrant, cos u = 4 ; what is tan a? Let Q(, ) be a point on the terminal arm of U that is r units from. cos U and cos U, so write and r 4 4 r Determine the value of. r Then P(, ) lies on the terminal arm of A., so tan A tan A, or b) The terminal arm lies in Quadrant 2, tan u = 7 ; what is sin a? Consider point Q from part a. tan U and tan U, so and 7 7 Determine the value of r. r 2 2 r r 8 Then P( 7, ) lies on the terminal arm of A. sin A, so sin A r 8 P D NT CPY.

8 c) The terminal arm lies in Quadrant 4, sin u = 2 ; what is cos a? Consider point Q from part a. sin U and sin U, so and r 2 2 r Determine the value of. r Then P( 9, ) lies on the terminal arm of A. 9 cos A r, so cos A 2 4. Which values of u satisf each equation for 0 u 60? a) tan u = b) tan u = 0 Use the table on page 444. U 4 and U 22 U 0, U 80, and U 60 c) sin u = d) sin u = 0 Use the table on page 444. U 90 U 0, U 80, and U 60 e) cos u = f) cos u = 0 Use the table on page 444. U 0 and U 60 U 90 and U 270. Which values of u satisf each equation for 0 u 60? a) tan u = b) sin u = Use the table on page 444. U and U U 270 c) cos u = Use the table on page 444. U To the nearest degree, which values of u satisf each equation for 0 u 60? a) tan u = b) tan u = 2 2 tan a 27 2 b tan a 2 b 4 tan U is positive in tan U is negative Quadrant, so U 27 or in in Quadrant 2, so Quadrant, so U 80 4, or 46 U 80 27, or 207 or in Quadrant 4, so U 60 4, or 26 6 D NT CPY. P

9 c) cos u = 0.6 d) sin u = 0.2 cos (0.6) sin (0.2) 4 cos U is positive in sin U is negative in Quadrant, so U or in Quadrant, so Quadrant 4, so U 80 4, or 94 U 60, or 07 or in Quadrant 4, so U 60 4, or a) Determine sin 40. For which values of u, where 0 u 60, is cos u = sin 40? sin Solve: cos U U cos ( ) U 0 cos U is also positive in Quadrant 4. U 60 0, or 0 b) Repeat part a for the sines of 4 different angles between 0 and 90. What patterns do ou see in the angles? Sample response: sin sin Solve: cos U Solve: cos U U 68 U Also, U 60 68, or 292 Also, U 60, or 29 sin sin Solve: cos U Solve: cos U U 4 U 7 Also, U 60 4, or 46 Also, U 60 7, or When the sine and the cosine of an angle are equal, the angles in Quadrant are complementar. 8. a) Determine tan 40. For which values of u, where 0 u 60, is tan u =? tan 40 tan Solve: tan U, or U tan (.97...) U 0 tan U is also positive in Quadrant. U 80 0, or 20 P D NT CPY. 7

10 b) Repeat part a for the tangents of 4 different angles between 0 and 90. What patterns do ou see in the angles? Sample response: tan tan Solve: Solve: tan U, or tan U, or U 68 U Also, U 80 68, or 248 Also, U 80, or 2 tan tan Solve: Solve: tan U, or tan U, or U 4 U 7 Also, U 80 4, or 94 Also, U 80 7, or 87 The tangents of complementar angles are reciprocals. 9. For each equation below: i) To the nearest degree, determine the possible values of u, for 0 u 60. ii) Determine the values of the other two primar trigonometric ratios of each angle u, to decimal places. a) cos u = 0.2 b) sin u = 0.9 i) cos (0.2) 78 i) sin (0.9) 64 cos U is positive in sin U is positive in Quadrant, so U 78 or in Quadrant, so U 64 or in Quadrant 4, so Quadrant 2, so U 60 78, or 282 U 80 64, or 6 ii) tan ii) tan tan tan sin cos sin cos c) tan u = 0.4 d) cos u = 0. i) tan (0.4) 24 i) cos (0.) 7 tan U is positive in cos U is negative in Quadrant, so U 24 Quadrant 2, so or in Quadrant, so U 80 7, or 07 U 80 24, or 204 or in Quadrant, so ii) cos U 80 7, or 2 cos ii) sin sin sin sin tan tan D NT CPY. P

11 e) sin u = 0. f) tan u =.8 i) sin (0.) 7 i) tan (.8) 6 sin U is negative in tan U is negative in Quadrant, so Quadrant 2, so U 80 7, or 97 U 80 6, or 9 or in Quadrant 4, so or in Quadrant 4, so U 60 7, or 4 U 60 6, or 299 ii) cos ii) sin cos sin tan cos tan cos C 20. a) For which values of u is tan u undefined, where 0 u 60? From the table on page 444, tan 90 and tan 270 are undefined. b) Are there an values of u, where 0 u 60, for which sin u or cos u are undefined? Justif our answers. sin U and cos U are defined for all values of U, where 0 U 60, because the are ratios with r in the denominator, and r To the nearest degree, which values of u satisf each equation for 0 u 60? a) tan u = tan 60 b) cos u = sin u tan U is negative in In Quadrant, where both Quadrants 2 and 4, so cos U and sin U are positive, U 80 60, or 20 ; U 4 and U 60 60, or 00 In Quadrant, where both cos U and sin U are negative, U 22 P D NT CPY. 9