SKILL BUILDER 5.2 Understanding Angles SKILL BUILDER A triangle has three angles and no angle can e equal to or greater than 18. Consider what happens when an angle is not part of a triangle ut is in the - plane. Angles and Their Location in the - Plane An angle is formed when a ra is rotated aout a fied point called the erte. The ra is called the at the eginning of the angle and the at the end of the angle. Angles are often laelled with Greek letters, such as theta, a alpha, and eta. An angle is in standard position if the erte of the angle is at the origin and the lies along the positie -ais. The can e anwhere on the arc of rotation. Standard Form Not Standard Form Not Standard Form erte erte erte An angle can e positie or negatie. A positie angle is formed a counterclockwise rotation of the. A negatie angle is formed a clockwise rotation of the. a positie angle a negatie angle The - plane is diided into four s the - and -aes. If is a positie angle, then the lies in I when < < 9 II when 9 < < 18 III when 18 < < 27 = 18 II III = 9 I IV = or 36 IV when 27 < < 36 = 27 418 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
Let P (, ) e a point on the of an angle in standard position. Since P can e anwhere in the - plane, the angle can terminate anwhere in the - plane. P(, ) 1 2 3 P(, ) 9 < 1 < 18 18 < 2 < 27 P (, ) lies in the negatie -ais. 1 terminates in II. 2 terminates in III. 3 27 Coterminal angles share the same and the same. As an eample, here are four different angles with the same and the same. P(, ) 1 2 3 4 If 1 12, then 2 36 12 3 72 12 4 36 12 48 84 24 The principal angle is the angle etween and 36. The coterminal angles of 48, 84, and 24 all share the same principal angle of 12. The related acute angle is the angle formed the of an angle in standard position and the -ais. The related acute angle is alwas positie and lies etween and 9. In this eample, represents the related acute angle for. 5.2 UNDERSTANDING ANGLES 419
Eample 1 Determine the principal angle and the related acute angle for 225. Solution Sketch 225 terminating in II. Lael the principal angle and the related acute angle. related acute angle = -225 principal angle The principal angle is the smallest positie angle that is coterminal to 225. In this case, 36 225 135. The related acute angle lies etween the and the -ais. It is positie ut less than 9. In this case, 225 ( 18 ) 45. Or, using the principal angle, 18 135 45. Eample 2 Determine the net two consecutie positie coterminal angles and the first negatie coterminal angle for 43. Solution Sketch each situation, showing the principal angle of 43. 43 43 43 (a) The first positie coterminal angle for 43 is 36 43 43. () The second coterminal angle is 36 36 43 763. (c) The first negatie coterminal angle is 36 43 317. Eample 3 Point P ( 3, 4) is on the of an angle in standard position. (a) Sketch the principal angle,. () Determine the alue of the related acute angle to the nearest degree. (c) Solution (a) What is the measure of to the nearest degree? Point P ( 3, 4) is in II, so the principal angle,, terminates in II. P(-3, 4) 4 2-4 -2 2 4 42 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
() The related acute angle,, is in the right triangle. P(-3, 4) 4 4 2-4 3 2 4 (c) The opposite side and the adjacent side are known so the tangent ratio can e used. tan o ad tan 4 3 pposite jacent tan 1 4 3 53 18 18 53 127 Sustitute known alues. Focus 5.2 Ke Ideas Angles can e located anwhere in the - plane. The - and -aes diide the - plane into four s. The erte of an angle in standard position is at the origin, and the of the angle is along the positie -ais. The of the angle can lie anwhere in the - plane. The of an angle rotates to its terminal position, either in a positie, counterclockwise direction or a negatie, clockwise direction. The principal angle is the first positie angle less than 36. The of an angle defines an infinite numer of coterminal angles. These can e positie or negatie and are defined in terms of the principal angle. The are multiples of 36 ; that is, 36 n, where n I. The related acute angle is the positie angle etween the and the -ais. It is alwas less than 9. An angle in standard position can e epressed in terms of its related acute angle. 5.2 UNDERSTANDING ANGLES 421
Practise, Appl, Sole 5.2 A 1. Sketch each angle in standard position. (a) 135 () 21 (c) 315 (d) 3 (e) 225 (f) 33 (g) 15 (h) 12 (i) 15 (j) 163 (k) 321 (l) 28 2. Determine the related acute angle for each angle in question 1. 3. Sketch each angle in standard position. (a) 379 () 491 (c) 545 (d) 64 (e) 593 4. Determine whether each pair of angles is coterminal or not. (a) 23, 383 () 41, 421 (c) 5, 31 (d) 38, 398 (e) 19, 39 (f) 41, 319 (g) 28, 232 (h) 15, 465 (i) 123, 237 (j) 19, 18 5. Calculate the net two positie coterminal angles. (a) 132 () 275 (c) 35 (d) 73 (e) 27 6. Calculate the net two negatie coterminal angles. (a) 53 () 138 (c) 299 (d) 18 (e) 192 7. Match each angle with its diagram. (a) 15 () 12 (c) 765 (d) 65 (e) 22 (f) 29 (g) 56 (h) 38 i. ii. iii. i.. i. 422 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
ii. iii. 8. Determine the principal angle. (a) 187 () 41 (c) 67 (d) 95 (e) 282 (f) 73 (g) 135 (h) 1249 9. State the principal angle for the gien related acute angle and gien. (a) 24, II () 35, III (c) 19, IV (d) 63, I B 1. State all alues of, where n I as shown. (a) 51 36 n, 4 n 6 () 71 36 n, 1 n 2 (c) 123 36 n, 2 n (d) 195 36 n, 5 n 7 11. Point P ( 9, 4) is on the of an angle in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? 12. Point P (7, 24) is on the of an angle in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? 13. Point P ( 5, 3) is on the of an angle,, in standard position. (a) Sketch the principal angle,. () What is the measure of the related acute angle to the nearest degree? (c) What is the measure of to the nearest degree? (d) What is the measure of the first negatie coterminal angle? 14. Check Your Understanding: Point P ( 5, 9) is on the of an angle in standard position. Eplain the role of the right triangle and the related acute angle in determining the principal alue of. C 15. Point P ( 5, 8) is on the of an angle,, in standard position. Determine all alues of for 54 27. 5.2 UNDERSTANDING ANGLES 423