additionalmathematicstrigonometricf unctionsadditionalmathematicstrigo nometricfunctionsadditionalmathem

Transcription

1 additionalmathematicstrigonometricf unctionsadditionalmathematicstrigo nometricfunctionsadditionalmathem TRIGNMETRIC FUNCTINS aticstrigonometricfunctionsadditiona lmathematicstrigonometricfunctions additionalmathematicstrigonometricf... unctionsadditionalmathematicstrigo nometricfunctionsadditionalmathem aticstrigonometricfunctionsadditiona lmathematicstrigonometricfunctions additionalmathematicstrigonometricf unctionsadditionalmathematicstrigo nometricfunctionsadditionalmathem aticstrigonometricfunctionsadditiona lmathematicstrigonometricfunctions additionalmathematicstrigonometricf unctionsadditionalmathematicstrigo nometricfunctionsadditionalmathem aticstrigonometricfunctionsadditiona lmathematicstrigonometricfunctions Name

2 TRIGNMETRIC FUNCTINS 5. Positive ngle and Negative ngle 90 Quadrant Quadrant II I 0 0 () Positive ngle Quadrant III Quadrant IV 3 70 ositive angle is measured in an anticlockwise direction from the ositive -ais. 360 () nticlockwise direction 60 radian 360 radian 0 Negative ngle negative angle is measured in a clockwise direction from the ositive -ais. 45 Clockwise direction Reresent each of the following angles in a Cartesian lane and state the quadrant of the angle. Eamle (a) 70 (b) 50 Eamle 5 Quadrant 5 (a) 95 (b) 345 Eamle 395 Quadrant III (a) 45 (b) 40 Eamle Quadrant I 5 4 4(a) 3 4 (b) 5 3 Eamle 45 Quadrant III 5(a) 30 (b) 3 45 Quadrant IV zefr@sas.edu.m

3 5. Si Trigonometric Functions of an ngle () 5.. Define sine, cosine and tangent of an angle in a Cartesian lane sin r cos Hotenuse osite to tan r 3 4 r r djacent to r Conclusion : Conclusion : sin cos tan Pthagoras Theorem : osite Hotenuse djacent Hotenuse osite djacent c a b c a b a c b a c b b c a b c a a b c 3. Find the length of and the values of sine, cosine and tangent of. (a) in quadrant I P (, 5) P sin 5 cos tan 5 5 (b) in quadrant II P (, 6) P sin 6 cos tan 6 6 zefr@sas.edu.m 3

4 (c) in quadrant III 4 3 P 3 sin 4 cos tan 3 P (4, 3) (d) in quadrant IV P sin cos 5 tan 5 P (5, ) (e) Conclusion: Sin is ositive for in quadrant. and. Cos is ositive for in quadrant. and. Tan is ositive for in quadrant. and. Sin is negative for in quadrant. and. Cos is negative for in quadrant. and. Tan is negative for in quadrant. and. 4. Find the corresonding reference angle of. 0 Fill in with or + sign. 90 Sin Cos Tan Sin Cos Tan 70 + Sin Cos Tan Sin Cos Tan (a) (b) Reference angle 55 Reference angle 70 0 (c) (d) Reference angle 5 35 zefr@sas.edu.m 4 Reference angle

5 (e) Conclusion: Reference angle (R) is the acute angle formed between the rotating ra of the angle and the R. R. R. R. In Quadrant II: In Quadrant III In Quadrant IV sin sin (0 ) sin sin ( 0) sin sin (360 ) cos cos (0 ) cos cos ( 0) cos cos (360 ) tan tan (0 ) tan tan ( 0) tan tan (360 ) Given that cos , find the trigonometric ratios of cos 3 without using a calculator or mathematical tables. Reference angle of 3 3 cos 3 6. Given that sin , find the trigonometric ratios of sin 60 without using a calculator or mathematical tables. Reference angle of sin Given that tan , find the trigonometric ratios of tan 335 without using a calculator or mathematical tables. Reference angle of tan 335 zefr@sas.edu.m 5

6 5.. Define cotangent, secant and cosecant of an angle in a Cartesian lane. sin cos tan r 3. Definition of cotangent, secant and cosecant. sin cosec cos sec tan cot r sin cos tan 4. Since tan cot sin cos, then r 5. r 90 sin sin 90 r cos r cos90 tan tan 90 r r 6. Comlementar angles: sin cos (90 ) cos sin (90 ) tan cot (90 ) cosec sec (90 ) sec cosec (90 ) cot tan (90 ) 7. Given that sin , cos and tan 4.06, evaluate the value of cos cos 4 4. Given that sin , cos and tan , evaluate the value of cot cot Given that sin , cos and tan , evaluate the value of sec sec zefr@sas.edu.m 6

7 5..3 Find values of si trigonometric functions of an angle. Comlete the table below sin cos cos ( ) cos sin ( ) sin tan ( ) tan tan. Use the values of trigonometric ratio for the secial angles, 30, 45 and 60, to find the value of the trigonometric functions below Eamle: Evaluate sin 0 Draw diagram to determine ositive or negative a. Evaluate tan 300 Draw diagram to determine ositive or negative sin Find reference angle Reference angle of Solve sin 0 sin 30 Find reference angle Solve b. Evaluate cos 50 c. Evaluate sec 35 Draw diagram to determine ositive or negative Draw diagram to determine ositive or negative Find reference angle Find reference angle Solve Solve zefr@sas.edu.m 7

8 5..4 Solve trigonometric equations. Stes to solve trigonometric equation. Determine the range of the angle.. Find the reference angle using tables or calculator. 3. Determine the quadrant where the angle of the trigonometric function is laced. 4. Determine the values of angles in the resective quadrants.. Solve the following equation for Eamle: sin 0.64 a. cos Range : Reference angle : sin Quadrant : S S S S T C T C T C T C Quadrant I Quadrant II Quadrant Quadrant ctual angles 40, 40, b. tan.9 c. cos Range : Reference angle : Quadrant : Quadrant Quadrant Quadrant Quadrant ctual angles zefr@sas.edu.m

9 d. sin e. tan Range : Reference angle : Quadrant : Quadrant Quadrant Quadrant Quadrant ctual angles f. cot.46 g. cosec.07 Range : Reference angle : Quadrant : Quadrant Quadrant Quadrant Quadrant ctual angles. Solve the following equation for eamle : sec a. sin.64 Range : cos Reference angle : cos 60 Quadrant : 0 S T C ,70 ctual angles 60, , , (36060) , 300, 40, 660 zefr@sas.edu.m 9

10 b. cos c. tan.05 Range Reference angle : Quadrant : ctual angles d. sin ( + 0) e. cos ( + 40) f. tan ( + 5) g. cos ( 0) 0.5 h. tan ( 0).0 i. sin ( 30) 0.5 zefr@sas.edu.m 0

11 j. sin cos 0 k. cos sin 55 Eamle : sin cos cos sin cos cos 0 cos ( sin ) 0 cos 0, sin 0 m. sin cos sin sin S T 90, 70 30, 50 30, 90, 50, 70 n. cos + 3 cos o. sin + 5 sin 3 C. tan tan q. 3 sin + cosec zefr@sas.edu.m

12 3. Given that sin and 0 0 < < Eress each of the following trigonometric ratios in terms of. (a) sec (b) cosec (c) tan (d) cot (e) sin ( ) (f) cos ( ) (g) sec ( ) (h) cosec (90 0 ) (i) tan ( 90 o - ) (j) cot ( 90 o ) (k) sin(-) (l) cos (-) zefr@sas.edu.m

13 4. Given that sin and 70 0 < < Without using tables or calculator, find the values of. 5. Given that cos - and 0 0 < < Without using tables or calculator, find the values of (a) cos (a) sin (b) tan (b) tan (c) cosec (c) cosec (d) sec (d) sec (e) cos (90 0 ) (e) sec (90 0 ) (f) sin ( 90 0 ) (f) cot ( 90 0 ) (g) sin (-) (g) sin (-) (h) tan (-) (h) cos (-) zefr@sas.edu.m 3

14 5.4 Basic Identities 5.4. Prove Trigonometric Identities using Basic Identities Three basic trigonometric identities : sin + cos + tan sec + cot cosec cos sin sin cos Formula of comound angle : sin ( B) sin cos B cos sin B cos ( B) cos cos B Ŧ sin sin B tan ( B) tan tan B tan tan B Formula of double angle : sin sin cos cos cos sin cos sin tan tan tan Formula of half angle : sin sin cos cos cos sin kos sin tan tan tan. Prove the following identities Eamle: cot + tan cosec sec cos sin cot tan sin cos cos sin sincos sincos cos ec sec a. tan ( sin ) sin b. sin cos cos c. sin + cot cosec cos zefr@sas.edu.m 4

15 sin e. sin d. sec tan sin cos sec cos sin. Solve the following equations for a. 3 sin + cosec b. cot 5 cot + 0 c. cos 3 sin d. cot + cosec e. tan 4 + sec zefr@sas.edu.m 5

16 NSWERS cos a. 0 70, , 6.3, 07.7, sin b ,.0 56, 64, 76, 4, 96, tan sin c. 0 0, 64 3 d. 40, 0. tan e. 5, 75 3(a) (b) (c) (d) (e) 5 5.(a) 7 (b) 5 (c) 7 5 (d) 7 5 (e) 7 9. cosec f. 30, 0 (f) 5..3 a. tan b. cos 50 3 g. 0, 30 h. 6.3, 5.3, 4.3, 33.3 (g) (h) (f) (g) (h) c. sec a.70, Quadrant I, IV 70, 90 b , 50, Quadrant I, IV 30, 330 c , 40, Quadrant II, III 40, 0 d , 70 Quadrant III, IV 50, 90 e , 0.0 Quadrant II, IV 59.99, f , 34 Quadrant II, IV 46, 36 g , 7 Quadrant III, IV 07, 333 i. 30, 90, 0, 70 j. 70, 0 k. 45, 5 (k) - m. 60, 0, 300 (l) n. 0, 0, 40 o. 30, 50. 0, 45, 5 q. 90, 99.47, (i) (j) 4.(a) 5 7 (b) 5 (c) 7 (d) (e) (f) (g) (h) zefr@sas.edu.m 6

17 7