Subject : Mathematics Topic : Trigonometric Ratio & Identity

Transcription

1 fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';ke iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks Vsd jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth 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. 0 Yrs. Que. from IEEE Student s Name : Class Roll No. : : DDRESS: R-, Opp. Raiway Track, New Corner Glass Building, Zone-, M.P. NGR, Bhopal : (0755) , , R TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 cos tan + sec tan sec + cos cos (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 cos θ 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 Prove that 40º cos 390º + cos ( 300º) ( 330º) tan 5º cot 405º + tan 765º cot 675º 0 TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 (, ] [, ) TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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) (f) cot ( ± B) tan ± tanb tan tanb cot B ± cot cot cot B TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

6 tan + tanb + tanc tan tanb tanc (g) tan ( + B + C) tan tanb tanb tan C tanc tan. Solved Example # 7 Prove that (45º + ) cos (45º B) + cos (45º + ) (45º B) cos ( B) 3 tan + θ tan + θ 4 4 Clearly (45º + ) cos (45º B) + cos (45º + ) (45º B) (45º º B) (90º + B) cos ( B) 6 of 9 TRIGONO METRIC RTIO & IDENTITY 3 tan + θ tan + θ 4 4 tan θ tanθ tanθ tan θ 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 TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 Prod uc ts into Sum or Dif 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 8θcos θ 6θcos3θ tan θ cosθcos θ 3θ4θ tan5θ + tan3θ 4 cos θ cos 4θ tan5θ tan3θ 8θcos θ 6θcos3θ cosθcos θ 3θ4θ 9θ + 7θ 9θ 3θ cos3θ + cos θ cos θ + cos7θ tan5θ + tan3θ tan5θ tan3θ θcos5θ cos5θcos θ 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 tan tan (d), cos tan tan 7 of 9 TRIGONO METRIC RTIO & IDENTITY TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 ; ; 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(b+c) 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. TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 x + 4cosx 5 3x + 4cosx 5 x + 3cos x 3 x + x + 4 x 3 x x Now 6 0 x x B 0 of 9 TRIGONO METRIC RTIO & IDENTITY TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

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 nswer max, min 4. Sine a nd Coe Series: α + (α + β) + (α + β ) ( α + β ) nβ n n β 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 4 6 cos + cos + cos nβ n β cos + cos + cos + cos + cos + cos (iii) cos + cos + cos + cos + cos 4 6 cos + cos + cos cos cos cos cos + cos + cos + cos + cos + cos n cos α + β TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL

12 cos cos (iii) cos + cos + cos + cos + cos 0 5 cos Find sum of the following series :. cos + cos n cos n to n terms. nswer n +. α + 3α + 4α nα, where (n + )α nswer 0. of 9 TRIGONO METRIC RTIO & IDENTITY TEKO CLSSES, H.O.D. MTHS : SUHG R. KRIY (S. R. K. Sir) PH: (0755) , , BHOPL