STUDY PACKAGE. Subject : Mathematics Topic : Trigonometric Ratio & Identity

Transcription

1 fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksMs rqjar e/;e eu dj ';ke iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks Vsd jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksmnklth egkjkt STUDY PCKGE Subject : Mathematics Topic : Trigonometric Ratio & Identity of 9 TRIGONO METRIC RTIO & IDENTITY Index. Theory. Short Revision 3. Exercise (Ex. to 5) 4. ssertion & Reason (Download Extra File) 5. Que. from Compt. Exams Yrs. Que. from IIT-JEE 7. 5 Yrs. Que. from IEEE Student s Name : Class Roll No. : : ddress : Plot No. 7, III- Floor, Near Patidar Studio, bove Bond Classes, Zone-, M.P. NGR, Bhopal : , , Whatspp R

2 Trigonometric Ratios & Identities. Basic Trigonometric Identities: (a) ² θ cos² θ ; θ ; cos θ θ R (b) sec² θ tan² θ ; sec θ θ R ( n ), n Ι (c) cosec² θ cot² θ ; cosec θ θ R { n, n Ι} Solved Example # Prove that cos 4 4 cos tan sec tan sec cos cos 4 4 (cos ) (cos ) cos [ cos ] cos tan sec tan sec tan sec (sec tan tan sec ) (tan sec )( sec tan ) tan sec tan sec cos Solved Example # If x x, then find the value of cos x 3 cos 0 x 3 cos 8 x cos 6 x cos x 3 cos 0 x 3 cos 8 x cos 6 x (cos 4 x cos x) 3 ( x x) 3 [ cos x x] 0 Solved Example # 3 If tan θ m, then show that sec θ tan θ m or 4m m Depending on quadrant in which θ falls, sec θ can be ± 4m 4m So, if sec θ 4m 4m m 4m

3 sec θ tan θ sec θ tan θ m and if sec θ m m 4m. Prove the followings : cos cos sec cosec (tan cot ) (iii) sec cosec tan cot (iv) (tan α cosec β) (cot β sec α) tan α cot β (cosec α sec β) (v) sec α cos α cosec α cos α α α αcos αcos α α 3 of 9 TRIGONO METRIC RTIO & IDENTITY m mn m mn. If θ, then prove that tan θ m mn n mn n. Circular Definition Of Trigonometric Functions: PM θ OP tan θ cot θ OM cos θ OP θ cos θ, cos θ 0 cos θ θ, θ 0 sec θ, cos θ 0 cosec θ, θ 0 cosθ θ 3. Trigonometric Functions Of llied ngles: If θ is any angle, then θ, 90 ± θ, 80 ± θ, 70 ± θ, 360 ± θ etc. are called LLIED NGLES. (a) ( θ) θ ; cos ( θ) cos θ (b) (90 θ) cos θ ; cos (90 θ) θ (c) (90 θ) cos θ ; cos (90 θ) θ (d) (80 θ) θ ; cos (80 θ) cos θ (e) (80 θ) θ ; cos (80 θ) cos θ (f) (70 θ) cos θ ; cos (70 θ) θ (g) (70 θ) cos θ ; cos (70 θ) θ (h) tan (90 θ) cot θ ; cot (90 θ) tan θ Solved Example # 4 Prove that cot tan (80º ) tan (90º ) tan (360º ) 0 sec (70º ) sec (90º ) tan (70º ) tan (90º ) 0 cot tan (80º ) tan (90º ) tan (360º ) cot tan cot tan 0 sec (70º ) sec (90º ) tan (70º ) tan (90º ) cosec cot 0 3. Prove that 40º cos 390º cos ( 300º) ( 330º) tan 5º cot 405º tan 765º cot 675º 0

4 4. Graphs of Trigonometric functions: (a) y x x R; y [, ] (b) y cos x x R; y [, ] 4 of 9 TRIGONO METRIC RTIO & IDENTITY (c) y tan x x R (n ) /, n Ι ; y R (d) y cot x x R n, n Ι; y R (e) y cosec x x R n, n Ι ; y (, ] [, ) (f) y sec x x R (n ) /, n Ι ; y (, ] [, )

5 Solved Example # 5 Find number of solutions of the equation cos x x 5 of 9 TRIGONO METRIC RTIO & IDENTITY Clearly graph of cos x & x intersect at two points. Hence no. of solutions is Solved Example # 6 Find range of y x x 3 x R We know x 0 x ( x ) 6 Hence range is y [, 6] 4xy 4. Show that the equation sec θ (x y) is only possible when x y 0 5. Find range of the followings. y x 5 x x R nswer [, 8] 3 y cos x cos x x R nswer, Find range of y x, x 3 nswer, 3 5. Trigonometric Functions of Sum or Difference of Two ngles: (a) ( ± B) cosb ± cos B (b) cos ( ± B) cos cosb B (c) ² ²B cos²b cos² (B). ( B) (d) cos² ²B cos²b ² cos (B). cos ( B) (e) tan ( ± B) tan ± tanb tan tanb cot cot B (f) cot ( ± B) cotb ± cot

6 tan tanb tanc tan tanb tan C (g) tan ( B C) tan tanb tanb tanc tanc tan. Solved Example # 7 Prove that (45º ) cos (45º B) cos (45º ) (45º B) cos ( B) tan θ tan θ Clearly (45º ) cos (45º B) cos (45º ) (45º B) (45º 45º B) (90º B) cos ( B) 6 of 9 TRIGONO METRIC RTIO & IDENTITY 3 tan θ tan θ 4 4 tanθ tanθ tanθ tanθ If α, cos β, then find (α β) nswer, Find the value of 05º nswer 9. Prove that tan tan tan cot sec 6. Factorisation of the Sum or Difference of Two Sines or C o s i n e s : C D (a) C D C D (c) cosc cosd cos Solved Example # 8 C D cos C D cos Prove that cos L.H.S cos [ C D Solved Example # 9 C D cos C D ] 3 C D (b) C D cos (d) cosc cosd C D R.H.S. Find the value of 3θ cos θ 4θ θ 3θ cos θ 4θ θ 3θ cos θ [ 3θ cos θ ] 0 C D C D

7 0. Proved that (iii) (iv) (v) cos 8x cos 5x 3x cos cos3 cos5 cos cos cos5 cos9 cos3 3x tan 4 cot 4 cos cos cot 7. Transformat io n of Produc ts into Sum or D if ference of S ines & C o s i n e s : (a) cosb (B) ( B) (b) cos B (B) ( B) (c) cos cosb cos(b) cos( B) (d) B cos( B) cos(b) Solved Example # 0 Prove that 8θcosθ 6θcos3θ tan θ cosθcos θ 3θ4θ tan5θ tan3θ 4 cos θ cos 4θ tan5θ tan3θ 8θcosθ 6θcos3θ cosθcosθ 3θ4θ 9θ 7θ 9θ 3θ cos3θ cosθ cos θ cos7θ tan5θ tan3θ tan5θ tan3θ θcos5θ cos5θcosθ 5θcos3θ 3θcos5θ 5θcos3θ 3θcos5θ θ 7θ 3θ θ. Prove that θ 5θ tan θ 8θ 4 cosθ cos 4θ θ. Prove that cos (B C) cos B (C ) cos C ( B) Prove that cos cos cos cos Multiple and Sub-multiple ngles : (a) cos ; θ θ cos θ (b) cos cos² ² cos² ²; cos² θ cos θ, ² θ cos θ. tan tan (c) tan ; tan θ tan tan θ θ 7 of 9 TRIGONO METRIC RTIO & IDENTITY tan (d) tan tan, cos tan

8 (e) (f) cos 3 4 cos 3 3 cos (g) tan 3 Solved Example # Prove that (iii) 3 tan tan 3 tan tan cos 3 tan cot cosec cos cosb cos( B) B tan cot cos cosb cos( B) L.H.S. cos L.H.S. tan cot (iii) L.H.S. cos tan cos tan cos tan tan cos cosb cos( B) cos cosb cos( B) B cos cos B cos tan cot B 4. Prove that θ θ tan θ cos θ cosθ B tan cos B 5. Prove that 0º 40º 60º 80º 6 3 tan tan 6. Prove that tan 3 tan tan tan 3 tan tan 7. Prove that tan 45 º sec tan 9. Important Trigonometric Ratios: B B cos B B (a) n 0 ; cos n ( ) n ; tan n 0, where n Ι cosec 8 of 9 TRIGONO METRIC RTIO & IDENTITY

9 (b) 5 or cos 5 or cos tan 5 5 (c) or cos 75 or cos or 3 3 cot 75 ; tan Conditional Identities: 5 & cos 36 or cos 5 4 ; ; 3 3 cot of 9 TRIGONO METRIC RTIO & IDENTITY If B C then : (iii) (iv) (v) B C 4 B C B C 4 cos cos B cos C cos cos B cos C 4 cos cos B cos C cos cos B cos C 4 B C tan tanb tanc tan tanb tanc (vi) tan tan B tan B tan C tan C tan (vii) cot cot B cot C cot. cot B. cot C (viii) cot cot B cot B cot C cot C cot (ix) B C Solved Example # then tan tan B tan B tan C tan C tan If B C 80, Prove that, B C cos cosb cosc.. Let S B C so that S cosb cosc cos(b C) cos(b C) cos cos(b C) cos(b C) S cos [cos(b C) cos(b C)] ce cos cos(bc) S cos cos B cos C Solved Example # 3 If x y z xyz, Prove that x x y y z z. Put x tan, y tanb and z tanc, so that we have tan tanb tanc tan tanb tanc Hence L.H.S. x x. y y. z z B C n, where n Ι.

10 x x y y z z tan tan tanb tan tan tanb tanc [ B C n ] tan tanb tanc x x y. y z. z 8. If B C 80, prove that (B C) (C ) ( B) 4 B C B C 8. B C B B C 9. If B C S, prove that (S ) (S B) S (S C) B. tanc. tan C C (S ) (S B) (S C) S 4 B C.. Range o f Trigonometric Expression: E a θ b cos θ b E a b (θ α), where tan α a a a b cos (θ β), where tan β b Hence for any real value of θ, a b E a b Solved Example # 4 Find maximum and minimum values of following : 3x 4cosx x 3cos x. We know 3 4 3x 4cosx 5 3x 4cosx 5 x 3cos x 3 x x 4 x 3 x Now 3 x x x B 0 of 9 TRIGONO METRIC RTIO & IDENTITY

11 3 3 3 x Find maximum and minimum values of following 3 (x ) nswer max, min 4. 0cos x 6x cosx x nswer max, min. (iii) cosθ 3 θ 6 nswer max, min 4. Sine a nd Coe Series: α (α β) (α β )... ( α β ) nβ n β n α β of 9 TRIGONO METRIC RTIO & IDENTITY cos α cos (α β) cos (α β )... cos ( α β ) Solved Example # 5 Find the summation of the following. cos 7 cos 7 4 cos 7 6 nβ n β cos cos cos cos cos cos (iii) cos cos cos cos cos cos 4 6 cos cos cos cos cos cos cos cos cos cos cos cos α n β

12 cos cos (iii) cos cos cos cos cos 0 5 cos 0 of 9 TRIGONO METRIC RTIO & IDENTITY Find sum of the following series :. cos cos n 3 cos n 5... to n terms. nswer n. α 3α 4α... nα, where (n )α nswer 0.