Radian Measure and Angles on the Cartesian Plane

. Radian Measure and Angles on the Cartesian Plane GOAL Use the Cartesian lane to evaluate the trigonometric ratios for angles between and. LEARN ABOUT the Math Recall that the secial triangles shown can be used to determine the eact values of the rimar and recirocal trigonometric ratios for some angles measured in degrees. B Q P? A 5 5 C How can these secial triangles be used to determine the eact values of the trigonometric ratios for angles eressed in radians? R EXAMPLE Connecting radians and the secial triangles Determine the radian measures of the angles in the secial triangles, and calculate their rimar trigonometric ratios. Solution /Q 5 5 5 a b 8 /B 5 /C 5 5 5 5 5 a b 8 5 /R 5 5 5 /P 5 /A 5 9 9 5 9 a b 8 5 a b 8 Q C A R P B ^PQR is the,, 9 secial triangle. Multil each angle b to 8 convert from degrees to radians. ^ABC is the 5, 5, 9 secial triangle. Multil each angle b to convert from 8 degrees to radians. NEL Chater

. P(, ) r = =. = sin 5 " cos 5 " csc 5 " sec 5 " tan 5 cot 5. P(, ). r = = =. sin 5 " cos 5 csc 5 " sec 5 tan 5 " cot 5 " Draw each secial angle on the Cartesian lane in standard osition. Use the trigonometric definitions of angles on the Cartesian lane to determine the eact value of each angle. Recall that sin u 5 r csc u 5 r cos u 5 r sec u 5 r. P(, ) r = =.. = sin 5 cos 5 " tan 5 " csc 5 sec 5 " cot 5 " tan u5 cot u5 where 5 r and r.. Reflecting A. Comare the eact values of the trigonometric ratios in each secial triangle when the angles are given in radians and when the angles are given in degrees. B. Elain wh the strateg that is used to determine the value of a trigonometric ratio for a given angle on the Cartesian lane is the same when the angle is eressed in radians and when the angle is eressed in degrees.. Radian Measure and Angles on the Cartesian Plane NEL

APPLY the Math. EXAMPLE Selecting a strateg to determine the eact value of a trigonometric ratio Determine the eact value of each trigonometric ratio. a) sin a b) cot a b b Solution a) P(, ) is one-quarter of a full revolution, and the oint P(, ) lies on the unit circle, as shown. Draw the angle in standard osition with its terminal arm on the ositive -ais. From the drawing, 5, 5, and r 5. sin a b 5 r 5 5 b) P(, ) is three-quarters of a full revolution, and the oint P(, ) lies on the unit circle, as shown. Draw the angle in standard osition with its terminal arm on the negative -ais. From the drawing, 5, 5 and r 5. cot a b 5 5 5 The relationshis between the rincial angle, its related acute angle, and the trigonometric ratios for angles in standard osition are the same when the angles are measured in radians and degrees. NEL Chater 5

EXAMPLE Selecting a strateg to determine the eact value of a trigonometric ratio Determine the eact value of each trigonometric ratio. a) cos a5 b) csc a b b Solution A: Using the secial angles a) 5 Sketch the angle in standard osition. is a 5 half of a revolution. is halfwa between and and lies in the third, quadrant with a related 5 angle of, or. = = r = P(, ) 5 cos a 5 b 5 r 5 " is in the,, " secial triangle. Position this triangle so the right angle lies on the negative -ais. Since (, ) lies on the terminal arm, 5, 5, and r 5 ". Therefore, the cosine ratio has a negative value. b) Sketch the angle in standard osition. is between and, and lies in the fourth quadrant with a related angle of, or.. Radian Measure and Angles on the Cartesian Plane NEL

= = r = P(, ) csc a b 5 r 5 5 is in the, ", secial triangle. Position it so that the right angle lies on the ositive -ais. Since the oint (", ) lies on the terminal arm, 5 ", 5, and r 5. Therefore, the csc ratio has a negative value.. Solution B: Using a calculator a) b) cos a 5 b 5 " Set the calculator to radian mode. Enter the eression. The result is a decimal. Entering confirms! that the answer is equivalent to this decimal. Tech Suort To ut a grahing calculator in radian mode, ress the MODE ke, scroll to Radian, and ress ENTER. There is no csc ke on the calculator. Use the fact that cosecant is the recirocal of sine. csc a b 5 NEL Chater 7

EXAMPLE If tan u5 7 where #u#, evaluate u to the nearest hundredth., Solution Solving a trigonometric equation that involves radians tan u 5 7 5 There are two ossibilities to consider: 5, 57and 5, 5 7. u P(, 7) For the ordered air (, 7), the terminal arm of the angle u lies in the fourth quadrant.,u,.8 8. In the fourth quadrant, u is about.. Use a calculator to determine the related acute angle b calculating 7 the inverse tan of. The related angle is.8, rounded to two decimal laces. Subtract.8 from to determine one measure of u. P(, 7) u.8 8.8 In the second quadrant, u is about.8. For the ordered air (, 7), the terminal arm of u lies in the second quadrant, and,u,, also has a related angle of.8. Subtract.8 from to determine the other measure of u. 8. Radian Measure and Angles on the Cartesian Plane NEL

In Summar Ke Ideas The angles in the secial triangles can be eressed in radians, as well as in degrees. The radian measures can be used to determine the eact values of the trigonometric ratios for multiles of these angles between and. The strategies that are used to determine the values of the trigonometric ratios when an angle is eressed in degrees on the Cartesian lane can also be used when the angle is eressed in radians.. The Secial Triangles The Secial Triangles on the Cartesian Plane Using a Circle of Radius P, Need to Know The trigonometric ratios for an rincial angle, u, in standard osition can be determined b finding the related acute angle, b, using coordinates of an oint that lies on the terminal arm of the angle. From the Pthagorean theorem, r 5, if r.. b u sin u 5 r csc u 5 r cos u 5 r sec u 5 r tan u 5 cot u 5 r P(, ) The CAST rule is an eas wa to remember which rimar trigonometric ratios are ositive in which quadrant. Since r is alwas ositive, the sign of each rimar ratio deends on the signs of the coordinates of the oint. In quadrant, All (A) ratios are ositive because both and are ositive. In quadrant, onl Sine (S) is ositive, since is negative and is ositive. In quadrant, onl Tangent (T) is ositive because both and are negative. In quadrant, onl Cosine (C) is ositive, since is ositive and is negative. S T A C NEL Chater 9

CHECK Your Understanding. For each trigonometric ratio, use a sketch to determine in which quadrant the terminal arm of the rincial angle lies, the value of the related acute angle, and the sign of the ratio. a) sin d) sec 5 b) cos 5 e) cos c) tan f) cot 7. Each of the following oints lies on the terminal arm of an angle in standard osition. i) Sketch each angle. ii) Determine the value of r. iii) Determine the rimar trigonometric ratios for the angle. iv) Calculate the radian value of u, to the nearest hundredth, where #u#. a) (, 8) c) (, ) b) (, 5) d) (, 5). Determine the rimar trigonometric ratios for each angle. 7 a) c) b) d). State an equivalent eression in terms of the related acute angle. a) sin 5 c) cot a b b) cos 5 d) sec 7 PRACTISING 5. Determine the eact value of each trigonometric ratio. K a) sin c) tan e) csc 5 b) cos 5 d) sin 7 f) sec 5. Radian Measure and Angles on the Cartesian Plane NEL

. For each of the following values of cos u, determine the radian value of u if #u#. a) c) " e) b) " d) " f) 7. The terminal arm of an angle in standard osition asses through each of the following oints. Find the radian value of the angle in the interval,, to the nearest hundredth. a) (7, 8) c) (, ) e) (9, ) b) (, ) d) (, ) f) (, ). 8. State an equivalent eression in terms of the related acute angle. a) cos c) csc a e) sin b b) tan d) cot f) sec 7 9. A leaning flagole, 5 m long, makes an obtuse angle with the ground. A If the distance from the ti of the flagole to the ground is. m, determine the radian measure of the obtuse angle, to the nearest hundredth.. The needle of a comass makes an angle of radians with the line ointing east from the centre of the comass. The ti of the needle is. cm below the line ointing west from the centre of the comass. How long is the needle, to the nearest hundredth of a centimetre?. A clock is showing the time as eactl :.m. and 5 s. Because a T full minute has not assed since :, the hour hand is ointing directl at the and the minute hand is ointing directl at the. If the ti of the second hand is directl below the ti of the hour hand, and if the length of the second hand is 9 cm, what is the length of the hour hand?. If ou are given an angle, u, that lies in the interval up c,, d C how would ou determine the values of the rimar trigonometric ratios for this angle?. You are given cos u5 5 where #u#., a) In which quadrant(s) could the terminal arm of u lie? b) Determine all the ossible trigonometric ratios for u. c) State all the ossible radian values of u, to the nearest hundredth. NEL Chater

. Use secial triangles to show that the equation cos Q 5 R 5 cos (5 ) is true. 5. Show that sin u 5 sin ucos u for. Determine the length of AB. Find the sine, cosine, and tangent ratios of /D, given AC 5 CD 5 8 cm. A. 8 cm B 8 cm C D 7. Given that is an acute angle, draw a diagram of both angles (in standard osition) in each of the following equalities. For each angle, indicate the related acute angle as well as the rincial angle. Then, referring to our drawings, elain wh each equalit is true. a) sin 5 sin ( ) c) cos 5cos ( ) b) sin 5sin ( ) d) tan 5 tan ( ) Etending 8. Find the sine of the angle formed b two ras that start at the origin of the Cartesian lane if one ra asses through the oint (!, ) and the other ra asses through the oint (,!). Round our answer to the nearest hundredth, if necessar. 9. Find the cosine of the angle formed b two ras that start at the origin of the Cartesian lane if one ra asses through the oint (!,!) and the other ra asses through the oint (7!, 7). Round our answer to the nearest hundredth, if necessar.. Julie noticed that the ranges of the sine and cosine functions go from to, inclusive. She then began to wonder about the recirocals of these functions that is, the cosecant and secant functions. What do ou think the ranges of these functions are? Wh?. The terminal arm of u is in the fourth quadrant. If cot u5!, then calculate sin u cot ucos u.. Radian Measure and Angles on the Cartesian Plane NEL