LEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot

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1 Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES a) state trignmetric ratis f sin,cs, tan, csec, sec and ct sin b) use tan =, sin(90 -) = cs, cs (90 -) = sin and cs tan(90 -) = ct c) use sme special angles d) evaluate trignmetric functins fr any angle e) use the Pythagrean identities : sin + cs = + ct = csec + tan = sec CONTENT P R θ Q sin cs ppsite hyptenuse adjacent hyptenuse ppsite tan adjacent PQ QR PQ PR QR PR

2 .a Trignmetric Ratis f sin,cs, tan, csec, sec and ct Mathematics DM 05 Tpic : Trignmetric Functins Fr any acute angle, there are six trignmetric ratis, each f which is defined by referring t a right angled triangle cntaining z y θ x Frm the diagram y sin csec = z sin = y z cs = z x tan = x y sec = ct = Cs = x z x = tan y Example 3 If ct, and is an acute angle, find tan,sin, sec and cs ec 4 Slutin

3 Mathematics DM 05 Tpic : Trignmetric Functins Example Given cs x = 0.8, evaluate 5 sin x 3 tan x + 3 csec x. Slutin Example 3 Given tan 7, find csec sec a) csec sec Slutin b) sin 4 cs 3

4 Mathematics DM 05 Tpic : Trignmetric Functins. b Relatinship f sin, cs and tan z y θ x Frm the diagram a) sin ( ) = z x cs = z x Therefre sin ( ) = cs b) cs ( ) = z y sin = z y Therefre cs ( ) = sin c) tan (90 0 x - ) = y Therefre tan ( ) = ct ct = tan y x = y x. c Trignmetric Ratis f Particular Angles Equilateral triangle f sides unit in length sin 60 0 = 3 sin 30 0 = 4

5 Mathematics DM 05 Tpic : Trignmetric Functins cs 60 0 = cs 30 0 = 3 tan 60 0 = 3 tan 30 0 = 3 Issceles triangle Hence, sin 45 0 =, cs 45 0 =, tan 45 0 = The values f trignmetric rati fr sme particular angles are as fllws: rad sin 0 cs tan rad rad rad rad 0 5

6 Mathematics DM 05 Tpic : Trignmetric Functins Example tan 30 Prve that cs60 0 tan 30 Slutin Example 5 Prve that sec 30 0 tan sin 45 0 csec cs 30 0 ct 60 0 = 7 Slutin 6

7 Mathematics DM 05 Tpic : Trignmetric Functins. d Trignmetric Rati fr Any Angle i) Psitive Angle First Quadrant y cs sin tan psitive value x Secnd Quadrant y sin (80 0 -) = sin cs (80 0 -) = - cs tan (80 0 -) = - tan x Third Quadrant y x sin ( ) = sin cs ( ) = - cs tan ( ) = tan 7

8 Mathematics DM 05 Tpic : Trignmetric Functins Furth Quadrant y sin ( ) = - sin cs ( ) = cs tan ( ) = -tan x We can summarize that all the trignmetric ratis are psitive in the first quadrant, sine is psitive in secnd quadrant, tangent is psitive in the third quadrant and csine is psitive in the furth quadrant as shwn belw. sin All tan cs ii) Negative Angle The rtating arm will describe a negative angle if it rtates in a clck-wise directin. T cnvert a negative angle t a nrmal basic angle, add r a multiple f The value f a trignmetrical rati f any negative angle can then be fund. Example An angle f is equivalent t a basic angle f =30 0. Then sin (-40 0 )= sin (30 0 )= - sin ( )= - sin 40 0 cs (-40 0 )= cs (30 0 )= cs ( )= cs 40 0 tan (-40 0 )= tan (30 0 )= - tan ( )= - tan 40 0 Hence in general sin (-) = -sin cs (-) = cs tan (-) = -tan 8

9 Mathematics DM 05 Tpic : Trignmetric Functins Example 6 State the trignmetric rati in acute angle. a) sin 60 0 b) cs 0 0 c) tan 30 0 d) cs 7 0 e) tan 46 0 Slutin Example 7 State the trignmetric rati in acute angle a) sin (-5 0 ) b) cs (-40 0 ) c) tan (-48 0 ) b) sin (-8 0 ) e) cs (-5 0 ) f) tan (-63 0 ) Slutin 9

10 Mathematics DM 05 Tpic : Trignmetric Functins Example 8 Find the fllwing a) sin 0 0 b) cs 0 0 c)tan 5 0 d) tan (-0 0 ) e) sin (-35 0 ) f) cs (-70 0 ) Slutin Example 9 Find the fllwing: a) sin b) cs c) tan 50 0 Slutin 0

11 Mathematics DM 05 Tpic : Trignmetric Functins. e Trignmetric Identities Y P ( x,y) r y O θ x X Fr any psitin f OP a right angled can be drawn fr which, x + y = OP = r Dividing thrugh by r becmes x ( ) r y r ( ) = Since x y = cs and sin r r Therefre cs + sin = Dividing thrught by x becmes Since y + ( ) x y x tan r ( ) x and r sec x Therefre + tan = sec Dividing thrugh by y becmes x ( ) y r ( ) y

12 Mathematics DM 05 Tpic : Trignmetric Functins Since x ct y and r y csec Therefre ct + = csec Useful frmulae and identities are summarized in the fllwing diagram. The diagnals shw the reciprcals f the varius trignmetric ratis: sin,cs, tan csec sec ct The shaded triangles shw the three basic identities. The sum f the square f the tw bases is equal t the square f the dwnward vertex. i.e sin + cs =, + tan = sec, + ct = csec sin cs gives by tan ct gives by sec csec

13 Mathematics DM 05 Tpic : Trignmetric Functins Example 0 Prve the identity tan + ct = sec csec Slutin Example Prve the identity ( sin + cs ) = ( sin )( + cs ) Slutin 3

14 Mathematics DM 05 Tpic : Trignmetric Functins Example sin sin Prve the identity Slutin tan sec 4

15 LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS Mathematics DM 05 Tpic : Trignmetric Functins SUBTOPIC :. COMPOUND ANGLE LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES a. use the frmulae sin A B, cs A B and A B b. use the duble-angle frmulae tan. a. use the frmulae sin A B, cs A B and A B tan. Example : Withut using calculatr, find the value fr the fllwing in terms f surd. sin a) b) cs c) tan Slutin: 5

16 Mathematics DM 05 Tpic : Trignmetric Functins Example : Express the fllwing in a single term. a) sin A csa csa sin A b) cs x cs 3 x cs x sin 3 x tan - c) tan Slutin: Example 3: Find the values f sin A B, cs A B, tan A B, if given: 3 5 a) sin A, csb, A and B in first quadrant b) csa, tan B, A in secndquadrant and Bin third quadrant. 3 4 Slutin: 6

17 Mathematics DM 05 Tpic : Trignmetric Functins Example 4: Prve the fllwing identities sin 60 A sin 60 a. - A sin A sin b. cs x y x cs y tan x tan y c. cs A cs B sin A sin B cs A B Slutin: 7

18 Mathematics DM 05 Tpic : Trignmetric Functins Exercise:. Withut using calculatr, find the value f: a) cs 75 - sin 75 b) tan 5 tan5 c) sin 73 cs3 - cs73 sin 3 Answer: - Answer: 3 Answer: 3. If A is an acute angle with the values f sin A B and 5 csa and B is an btuse angle with csb, find 7 5 cs A B 9 33 Answer: -, Simplify this fllwing a) sin A csa csa sin A Answer: sin 3A b) tan - tan tan Prve that tan Answer: 8

19 Mathematics DM 05 Tpic : Trignmetric Functins 5.Prve each f given identies. sin A B sin A - B sin sin A - B b. tan A - tan B csa cs B ct A ct B - c. ct A B ct A ct B a. A sin B 6. Given sin x y pcsx - y, shw that p - tan y tan x - p tan y b. use the duble-angle frmulae Example 5: Find the exact value f Slutin: tan5 - tan 5 Example 6: 3 If tan with is acute angle, find the exact value f: 4 a) tan b) tan 4 Slutin: 9

20 Mathematics DM 05 Tpic : Trignmetric Functins Example 7: Withut using calculatr find the exact value f; a) sin.5 cs.5 b) - sin 75 tan 5 c) - tan 5 Slutin: Example 8: Prve each f the fllwing: a) ct A tan A ct A sec A 3 b) sin3a sin A - 4 sin A Slutin: 0

21 Mathematics DM 05 Tpic : Trignmetric Functins Example 9: Given that is acute and that tan, evaluate each f the fllwing; a) tan b) sin c) sec Slutin:

22 Mathematics DM 05 Tpic : Trignmetric Functins Exercise:. Simplify the expressins a. sin A cs A b. tan Acs A c. sin A csa Answer: a. sin A b. sin A c. 5. Given that is an acute angle and that cs, find the value f each f these. 3 a. cs b. csec c. ct Answer: a b c If sin C find sinc, cscand tan C if 5 a. C is acute b. C is btuse Answer: a.,, b.,, Prve f the given identities. tan d. sin tan csa sin A cs A B e. sin B csb sin B 4 4 f. cs sin cs cs g. cs sin cs sin cs 5. Find an expressin fr Answer: a, b tan in the frm a b, where a and b are integers.

23 Mathematics DM 05 Tpic : Trignmetric Functins LECTURE 3 OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :.3 COMPOUND ANGLE LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES c) use the half-angle frmulae d) use the factr frmulae A B A - B i. sin A sin B sin cs A B A - B ii. sin A sin B cs sin A B A - B iii. csa csb cs cs A B A - B iv. csa csb sin sin c. use the half-angle frmulae Example : Given that sin A = and cs B = 5 3 where A and B are in the same quadrant, find withut using calculatrs (a) sin (A + B) (b) cs A (c) sin A Slutin : 3

24 Mathematics DM 05 Tpic : Trignmetric Functins Example : x tan x sin x Shw that sin tan x Slutin: 4

25 Mathematics DM 05 Tpic : Trignmetric Functins d. use the factr frmulae Example 3: Express each f the fllwing as the prduct f tw trignmetric functins. a. sin 6 sin 4 b. cs8 cs4 c. cs7 cs5 d. sin 8 sin Slutin: Example 4: Express each f the fllwing as the sum r difference f tw trignmetric functins. a. sin 6A sin A b. sin 3A csa c. cs 4A sin A d. 4 cs5a cs3a Slutin: 5

26 Mathematics DM 05 Tpic : Trignmetric Functins Example 5: Find the exact value withut using calculatr a) sin 05 sin 5 b) sin 05 - sin 5 sin 75 sin5 c) cs 75 cs5 d) cs 5 - sin 5 Slutin: 6

27 Mathematics DM 05 Tpic : Trignmetric Functins Example 6: Prve the fllwing identities sin 7A sin 3A a) tan 5A cs7a cs3a csb- csa tan A - B sin A csb 6 c) sin 75 sin 50 b) Slutin : 7

28 Mathematics DM 05 Tpic : Trignmetric Functins Exercise:. Express each f the fllwing prducts a sum cntaining nly sines r csines (a) sin 6θ sin 4θ (b) cs 3θ cs θ (c) sin 3θ cs 5θ Answer: a) (cs - cs0 ) b) (cs4θ + cs θ ) c) (sin 8θ sin θ ). Express each sum r difference as a prduct f sines and / r csines (a) sin 5θ - sin 3θ (b) cs 3θ + cs θ 5 Answer: a) cs 4 θ sin θ b) cs cs 3. Find the exact value withut using calculatr (a) 75 sin5 cs A cs 0 Answer: (a) (b) sin (b) - A cs 0 A 4. Prve the fllwing identities (a) cs θ + cs 3θ + cs 4θ cs 3θ ( + cs θ) sin 3A sin 6A sin A sin A (b) tan 5A sin 3A cs6a sin A csa sin 3A - sin A (c) - ct A cs3a csa 8

29 Mathematics DM 05 Tpic : Trignmetric Functins LECTURE 4 OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :.3 Slutin f Trignmetric Equatins LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES a) slve equatins such as sin = k, cs = k, tan = k. b) slve equatins in quadratic frm a. Slutin f Trignmetric Equatins Slve Equatins such as sin = k, cs = k, tan = k. Example Find the value f, if a) sin = b) cs = c) sin = d) tan = - e) cs = 4 f) sin = g ) cs = sin 85 0 h) tan = 3 4 Slutin 9

30 Mathematics DM 05 Tpic : Trignmetric Functins Example : Slve the fllwing equatins fr angles in the range a) tan = 3 b) sin = c) cs (3x ) = 0.5 d) cs x 0. 3 Slutin 30

31 Mathematics DM 05 Tpic : Trignmetric Functins Example 3: Slve the equatins Slutin tan sin in radian, where 0 3

32 Mathematics DM 05 Tpic : Trignmetric Functins b Slve Equatins In Quadratic Frm In rder t slve trignmetric equatins such as sin - cs = 0 and 4 cs + 3 sin = 4 we will reduce them t ne r mre f the frms sin = k, cs = k r tan = k (where k is a cnstant), by using ne r mre f the trignmetric identities. Example 4 Slve the fllwing trignmetric equatins a) 6 cs x + cs x = x 360 b) sin x sin x - = x 360 c) ct + 5 csec + = 0 0 d) 4 cs x = 3 sin x x 360 e) sin + 5 cs + = Slutin 3

33 33 Mathematics DM 05 Tpic : Trignmetric Functins

34 Mathematics DM 05 Tpic : Trignmetric Functins LECTURE 5 OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :.3 Slutin f Trignmetric Equatins LEARNING : OUTCOMES At the end f the lessn, students shuld be able t: c) express sin, cs and tan in terms f t where d) express a cs ± b sin as R cs ( e) slve a cs + b sin = c θ t tan ) r R sin f) determine the maximum and minimum values f trignmetric expressins c. Express sin, cs & tan in Terms f t where t = tan tan = tan tan tan = tan tan by using t = tan t tan = t t sin = t t cs = t t - t + t Nte Equatins in the frm a cs + b sin = k can be slved by using the expressins abve. 34

35 Example Slve the equatins 3 cs x 8 sin x = -, Mathematics DM 05 Tpic : Trignmetric Functins 0 0 by using the t-substitutin Slutin 35

36 Example Slve the equatins 5 tan + sec + 5 = 0 fr values f in t = tan Mathematics DM 05 Tpic : Trignmetric Functins by using Slutin 36

37 d. Express a cs b sin as R cs ( ) r R sin ( ) Mathematics DM 05 Tpic : Trignmetric Functins The equatin 3 cs + sin = can als be slved if the expressin 3 cs + sin is 0 expressed in the frm R cs ( - ), where R > 0 and (0 < <90 0 ) are values t be determined. Let R cs ( - ) = a cs + b sin R (cs cs + sin sin ) = a cs + b sin (R cs ) cs + (R sin ) sin = a cs + b sin Equating the cefficient f cs : R cs = a [] Equating the cefficient f sin : R sin = b [] Squaring each f [] and [] and adding give [] + [] : R cs + R sin = a + b R (cs + sin ) = a + b R = a + b R a b [] [] : R sin R cs tan b a b a Hence, the expressin a cs + b sin can be expressed in the frm R cs ( ), b where R a b and tan a We cnclude that : a cs b sin R cs a sin b cs R sin where where R R a a b b and and b tan a a tan b 37

38 Mathematics DM 05 Tpic : Trignmetric Functins Example 3 Express 3 cs - 4 sin in the frm R cs ( + ) Slutin 38

39 Mathematics DM 05 Tpic : Trignmetric Functins e. slve a cs + b sin = c Example 4 Slve the fllwing equatins fr Slutin sin - 3 cs = 3 39

40 Example 5 Slve the fllwing equatins fr Slutin: cs + 3 sin = Mathematics DM 05 Tpic : Trignmetric Functins 40

41 f. Maximum and Minimum Values f Trignmetric Expressin Mathematics DM 05 Tpic : Trignmetric Functins a sin x + b cs x can be rewritten using an expressin f the frm R sin (x ) r R cs (x ) where is the phase angle and R is the amplitude & depends n the values a and b. sine wave sin x csine wave cs x Maximum value: sin 90 0 = cs 0 and = Minimum value: sin 3 = - cs = - 4

42 Mathematics DM 05 Tpic : Trignmetric Functins Example 6 Find the maximum & minimum values fr the expressin 3 sin x 4 cs x and find the values f x, where 0 < x < 360, fr which the expressin has the extreme values. Slutin: 4

43 Mathematics DM 05 Tpic : Trignmetric Functins Example 7 Find the maximum & minimum values fr the expressin f. 3 sin - cs Slutin: 43

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