SHORT REVISION. FREE Download Study Package from website: 2 5π (c)sin 15 or sin = = cos 75 or cos ; 12

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1 SHORT REVISION Trigoometric Rtios & Idetities BASIC TRIGONOMETRIC IDENTITIES : ()si θ + cos θ ; si θ ; cos θ θ R (b)sec θ t θ ; sec θ θ R (c)cosec θ cot θ ; cosec θ θ R IMPORTANT T RATIOS: ()si π 0 ; cos π (-) ; t π 0 where I ( + ) π ( + ) π (b)si ( ) &cos 0 where I π 5π (c)si 5 or si cos 75 or cos ; π + 5π cos 5 or cos si 75 or si ; + t 5 cot 75 ; t 75 + cot 5 + π + ; cos π (d)si π 5 (e) si or si 0 π π ; t ; t + π 5+ & cos or cos 5 TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES : If θ is y gle, the θ, 90 ± θ, 0 ± θ, 70 ± θ, 0 ± θ etc re clled ALLIED ANGLES () si ( θ) si θ ; cos ( θ) cos θ (b) si (90 - θ) cos θ ; cos (90 θ) si θ (c) si (90 + θ) cos θ ; cos (90 + θ) si θ (d)si (0 θ) si θ; cos (0 θ) cos θ (e) si (0 + θ) si θ ; cos (0 + θ) cos θ (f) si (70 θ) cos θ ; cos (70 θ) si θ (g) si (70 + θ) cos θ ; cos (70 + θ) si θ TRIGONOMETRIC FUNCTIONS OF SUM OR DIFFERENCE OF TWO ANGLES : () si (A ± B) sia cosb ± cosa sib (b) cos (A ± B) cosa cosb sia sib (c) si²a si²b cos²b cos²a si (A+B) si (A B) (d) cos²a si²b cos²b si²a cos (A+B) cos (A B) t A ± t B cot A cot B (e) t (A ± B) (f) cot (A ± B) t A t B cot B ± cot A 5 FACTORISATION OF THE SUM OR DIFFERENCE OF TWO SINES OR COSINES : C+ D C D C+D C D () sic + sid si cos (b) sic sid cos si C+D C D C+ D C D (c) cosc + cosd cos cos (d) cosc cosd si si TRANSFORMATION OF PRODUCTS INTO SUM OR DIFFERENCE OF SINES & COSINES : () sia cosb si(a+b) + si(a B) (b) cosa sib si(a+b) si(a B) (c) cosa cosb cos(a+b) + cos(a B) (d) sia sib cos(a B) cos(a+b) 7 MULTIPLE ANGLES AND HALF ANGLES : of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL () si A sia cosa ; si θ si θ cos θ

2 (b) cosa cos A si A cos A si A ; cos θ cos θ si² θ θ cos si θ cos A cos A + cos A, si A cos A ; t A cos A cos θ + cos θ, si θ cos θ ta t( θ ) (c) t A ; t θ t A t ( θ ) ta t A (d) si A, cos A (e) si A sia si A t A t A (f) cos A cos ta t A A cosa (g) t A t A THREE ANGLES : ta+ tb+ tc tatbtc () t (A+B+C) tatb tbtc tcta NOTE IF : (i) A+B+C π the ta + tb + tc ta tb tc π (ii) A+B+C the ta tb + tb tc + tc ta (b) If A + B + C π the : (i) sia + sib + sic sia sib sic A B C (ii) sia + sib + sic cos cos cos 9 MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS: () Mi vlue of t θ + b cot θ b where θ R (b) Mx d Mi vlue of cosθ + bsiθ re + b d + b (c) If f(θ) cos(α + θ) + bcos(β + θ) where, b, α d β re kow qutities the + b + b cos( α β) < f(θ) < + b + bcos( α β) (d) If α,β 0, π d α + β σ (costt) the the mximum vlues of the expressio cosα cosβ, cosα + cosβ, siα + siβ d siα siβ occurs whe α β σ/ (e) If α,β 0, π d α + β σ(costt) the the miimum vlues of the expressio secα + secβ, tα + tβ, cosecα + cosecβ occurs whe α β σ/ (f) If A, B, C re the gles of trigle the mximum vlue of sia + sib + sic d sia sib sic occurs whe A B C 0 0 (g) I cse qudrtic i siθ or cosθ is give the the mximum or miimum vlues c be iterpreted by mkig perfect squre 0 Sum of sies or cosies of gles, si α + si (α + β) + si (α + β ) + + si ( α + β) cos α + cos (α + β) + cos (α + β ) + + cos ( α + β) si si EXERCISE I β β si si si α+ β β β cos α+ β Q Prove tht cos²α + cos² (α + β) cos α cos β cos (α + β) si²β Q Prove tht cos α si²β + cos (α + β) si α si β + cos (α + β) Q Prove tht, t α + t α + t α + cot α cot α Q Prove tht : () t 0 t 0 t 0 t 0 π π 5π 7π (b) t 9 t 7 t + t (c) si + si + si + si Q5 Clculte without usig trigoometric tbles : cos 0 cos0 () cosec 0 sec 0 (b) cos 0 cot 0 (c) si 0 sec5 cos0 π π 5π 7π (d) si0 + si5 (e) cos + cos + cos + cos si5 (f) t 0 t 50 + t 70 7π Q() If X si θ + π π 7π + si θ + si θ +, Y cos θ + π π + cos θ + cos θ + of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

3 X Y the prove tht tθ Y X (b) Prove tht si² + si² + si² 9 + si² si² 9 + si² Q7 Show tht : () cot 7 or t ( + )( + ) or (b) t + m+ Q If m t (θ - 0 ) t (θ + 0 ), show tht cos θ (m ) π y Q9 If t + t π x si y + si x +, prove tht si x si x Q0 If cos (α + β) 5 ; si (α - β) 5 & α, β lie betwee 0 & π, the fid the vlue of t α tβ siβ Q Prove tht if the gles α & β stisfy the reltio ( m > ) t α tα tβ the si( α+β) m m+ m Q () If y 0 cos²x si x cos x + si²x, the fid the gretest & lest vlue of y (b) If y + si x + cos x, fid the mximum & miimum vlues of y x R (c) If y 9 sec x + cosec x, fid the miimum vlue of y x R (d) Prove tht cos θ (e) Prove tht ( ) π cos θ + lies from - & 0 + si θ + cos θ lies betwee ( + 5) & ( 5) + ta Q If A + B + C π, prove tht (t A) (cot A) tbtc Q If α + β c where α, β > 0 ech lyig betwee 0 d π/ d c is costt, fid the mximum or miimum vlue of () si α + si β (b) si α si β (c) t α + t β (d) cosec α + cosec β Q5 Let A, A,, A be the vertices of -sided regulr polygo such tht ; + Fid the vlue of A A A A A A Q Prove tht : cosec θ + cosec θ + cosec θ + + cosec θ cot (θ/) cot - θ Q7 For ll vlues of α, β, γ prove tht; α+β β+ γ γ+α cos α + cos β + cos γ + cos (α + β + γ) cos cos cos si A cos B si A si B Q Show tht + cos A si B si(a B) + cos A cos B t α + t γ si α + si γ Q9 If t β, prove tht si β + t α t γ + si α si γ Q0 If α + β γ, prove tht cos² α + cos² β + cos² γ + cos α cos β cos γ β γ ( )( )( ), show tht Q If α + β + γ π t α t t β γ ( t α )( t )( t ) Q If A + B + C π d cot θ cot A + cot B + cot C, show tht, si (A θ) si (B θ) si (C θ) si θ π π 5π 7π Q If P cos + cos + cos + + cos d π π π 0π Q cos + cos + cos + + cos, the fid P Q si α + siβ + si γ cosα + cosβ + cos γ Q If A, B, C deote the gles of trigle ABC the prove tht the trigle is right gled if d oly if sia + sib + sic 0 Q5 Give tht ( + t )( + t )( + t 5 ), fid EXERCISE II Q If t α p/q where α β, α beig cute gle, prove tht; Q (p cosec β q sec β) p + q Let A, A, A A re the vertices of regulr sided polygo iscribed i circle of rdius R If (A A ) + (A A ) + + (A A ) R, fid the umber of sides i the polygo of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

4 cosθ + cosφ Q Prove tht: (cosθ + cosφ) cos(θ + φ) (siθ + siφ) si(θ + φ) cos( θ φ) Q Without usig the surd vlue for si 0 or cos 0, prove tht si 0 cos 0 5 six six si9x Q5 Show tht, + + (t7x tx) cosx cos9x cos7x 5 5 r π r π Q Let x cos d x r cos, the show tht r π x x cos ec, where Π deotes the cotiued product π Q7 If θ, prove tht t θ t θ + t θ t θ + t θ t θ 7 7 Q For 0 < x < π cosx prove tht, > si x(cosx six) π 7 π π π Q9 () If α prove tht, si α + si α + si α (b) si si si cosk Q0 Let k, the prove tht cos k cos( ) k si k 0 + Q Prove tht the vlue of cos A + cos B + cos C lies betwee & where A + B + C π Q If cosa tb, cosb tc d cosc ta, the prove tht sia sib sic si + cos x Q Show tht x R c ot hve y vlue betwee d Wht iferece si x si x c you drw bout the vlues of? + cos x Q If ( + si t)( + cos t) 5 Fid the vlue of ( si t)( cos t) si α cos α si α cos Q5 Prove tht from the equlity + follows the reltio ; + b + b b Q Prove tht the trigle ABC is equilterl iff, cot A + cot B + cot C Q7 Prove tht the verge of the umbers si,,,,, 0, is cot Q Prove tht : si 7 ( 5+ 5) / ( 5) / A Q9 If A+B+C π; prove tht t B C + t + t Q0 If A+B+C π (A, B, C > 0), prove tht si A si B si C α 7 ( + b) Q Show tht ellimitig x & y from the equtios, si x + si y ; b cos x + cos y b & t x + t y c gives c ( + b ) Q Determie the smllest positive vlue of x (i degrees) for which t(x + 00 ) t(x + 50 ) t x t (x 50 ) x t Q Evlute : x cos β + γ α γ + α β α + β γ Q If α + β + γ π & t t t, the prove tht; + cos α + cos β + cos γ 0 Q5 x R, fid the rge of the fuctio, f (x) cos x (si x + si x + si α ) ; α [0, π] EXERCISE III Q sec xy θ is true if d oly if : [JEE 9, ] (x+ y) (A) x + y 0 (B) x y, x 0 (C) x y (D) x 0, y 0 Q () Let be odd iteger If si θ (A) b 0, b r 0 b r si r θ, for every vlue of θ, the : (B) b 0 0, b 5 of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

5 (C) b 0, b (D) b 0 0, b + (b) Let A 0 A A A A A 5 be regulr hexgo iscribed i circle of uit rdius The the product of the legths of the lie segmets A 0 A, A 0 A & A 0 A is : (A) (B) (C) (D) (c) Which of the followig umber(s) is/re rtiol? [ JEE '9, + + out of 00 ] (A) si 5º (B) cos 5º (C) si 5º cos 5º (D) si 5º cos 75º θ Q For positive iteger, let f (θ) t ( sec θ) ( sec θ) ( sec θ) ( + sec θ) The π π π π (A) f (B) f (C) f (D) f 5 [JEE '99,] Q() Let f (θ) si θ (si θ + si θ) The f (θ) : [ JEE 000 Screeig out of 5 ] (A) 0 oly whe θ 0 (B) 0 for ll rel θ (C) 0 for ll rel θ (D) 0 oly whe θ 0 (b) I y trigle ABC, prove tht, cot A + cot B + cot C cot A cot B cot C [JEE 000] Q5() Fid the mximum d miimum vlues of 7 cos x si x (b) Fid the smllest positive vlues of x & y stisfyig, x y π, cot x + cot y [REE 000, ] Q If α + β π d β + γ α the tα equls [ JEE 00 (Screeig), out of 5 ] (A) (tβ + tγ) (B) tβ + tγ (C) tβ + tγ (D) tβ + tγ Q7 If θ d φ re cute gles stisfyig siθ, cos φ, the θ + φ [JEE 00 (Screeig)] π π π π π 5π 5π (A), (B), (C), (D), π Q I equilterl trigle, cois of rdii uit ech re kept so tht they touch ech other d lso the sides of the trigle Are of the trigle is (A) + (B) + (C) + 7 (D) + 7 [JEE 005 (Screeig)] π Q9 Let θ 0, d t (tθ) tθ, t (tθ) cotθ, t (cotθ) tθ, t (cotθ) cotθ, the (A) t > t > t > t (B) t > t > t > t (C) t > t > t > t (D) t > t > t > t [JEE 00, ] ANSWER SHEET (EXERCISE I) Q 5 () (b) (c) (d) (e) 5 (f) Q 0 5 Q () y mx ; y mi (b) y mx ; y mi, (c) 9 Q () mx si (c/), (b) mx si (c/), (c) mi t (c/), (d) mi cosec (c/) Q 5 7 Q Q5 EXERCISE II Q 7 Q, Q 0 Q x 0 Q Q5 si x x si α y si α si EXERCISE III Q B Q () B, (b) C, (c) C Q A, B, C, D Q () C Q5 () mx 5 & mi 5 5π π ; (b) x ; y Q C Q7 B Q B Q9 B of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

6 EXERCISE IV (Objective) Prt : (A) Oly oe correct optio t ( x )cos ( π π + x) si ( 7π x) cos( x π )t ( π + x) whe simplified reduces to: (A) si x cos x (B) si x (C) si x cos x (D) si x π The expressio + π + α π si α si ( ) + π + α si + α si (5 ) is equl to (A) 0 (B) (C) (D) si α + si α If t A & t B re the roots of the qudrtic equtio x x + b 0, the the vlue of si (A + B) (A) (B) (C) (D) + ( b) + b (b+ c) b ( ) The vlue of log [cos (α + β) + cos (α β) cos α cos β] : (A) depeds o α & β both (B) depeds o α but ot o β (C) depeds o β but ot o α (D) idepedet of both α & β cos0 + si70 si50 si0 5 si 0 is equl to: (A) (B) (C) / (D) oe If cos A /, the the vlue of cos (A/) si (A/) si (5A/) is 7 (A) (B) (C) (D) If y cos (5º + x) + (si x cos x) the the mximum & miimum vlues of y re: (A) & 0 (B) & 0 (C) & (D) oe π The vlue of cos 9 π 5π 7π + cos + cos + + cos is equl to: (A) / (B) 0 (C) (D) oe 9 The gretest d lest vlue of log ( six cosx + ) re respectively: (A) & (B) 5 & (C) 7 & 5 (D) 9 & 7 0 I right gled trigle the hypoteuse is times the perpediculr drw from the opposite vertex The the other cute gles of the trigle re π π π π π π π π (A) & (B) & (C) & (D) & 5 0 cos90 (A) + si50 (B) (C) (D) oe π If < α < π, the cot α + is equl to si α (A) + cot α (B) cot α (C) cot α (D) + cot α π π x If x π, the cos + si x + si x is lwys equl to (A) (B) (C) (D) oe of these If cos x + si x, the vlue of 7 cos x + si x is equl to (A) or (B) or (C) or (D) oe of these 5 If cosec A + cot A, the t A is (A) If cot α + t α m d 5 (B) cos α cos α, the (C) 7 7 (D) (A) m (m ) / (m ) / (B) m(m ) / (m ) / (C) (m ) / m(m ) / (D) (m ) / m(m ) / 7 cosx + cos x + 5cosx + 0 The expressio is equl to cos5x + 5cos x + 0cos x (A) cos x (B) cos x (C) cos x (D) + cos x If sia sib cos A d cosb 5, 0 < A, B < π/, the t A + t B is equl to (A) / 5 (B) 5 / (C) (D) ( 5 + )/ 5 9 If si θ k, the the vlue of k (A) k (B) k k t t θ θ + cot cot θ θ is equl to (C) k + (D) k 7 of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

7 Prt : (B) My hve more th oe optios correct 0 Which of the followig is correct? (A) si > si (B) si < si (C) cos > cos (D) cos < cos If si β si (α + β), the t (α + β) t α is (A) idepedet of α (B) idepedet of β (C) depedet of both α d β (D) idepedet of α but depedet of β si( α + β) π cos ( α + β) cos It is kow tht si β & 0 < β < π the the vlue of is: 5 siα (A) idepedet of α for ll β i (0, π) (B) 5 for t β > 0 (7 + cot α) (C) for t β < 0 (D) oe 5 If the sides of right gled trigle re {cosα + cosβ + cos(α + β)} d {siα + siβ + si(α + β)}, the the legth of the hypoteuse is: α β α + β (A) [cos(α β)] (B) [ cos(α + β)] (C) cos (D) si If x sec φ t φ & y cosec φ + cot φ the: y + + x y (A) x y (B) y x (C) x y + (D) xy + x y ( + ) si α + ( ) cos α ( + ) if t α (A) (B) (C) (D) + b If t x, ( c) c y cos x + b si x cos x + c si x z si x b si x cos x + c cos x, the (A) y z (B) y + z + c (C) y z c (D) y z ( c) + b cosa + cosb sia + sib 7 + sia sib cosa cosb A B A B (A) t (B) cot : is eve (C) 0 : is odd (D) oe The equtio si x + cos x hs rel solutio if (A) (, ) (B), (C) (D), EXERCISE IV (Subjective) The miute hd of wtch is 5 cm log How fr does its tip move i 50 miutes? (Use π ) If the rcs of the sme legth i two circles subted gles 75 d 0 t the cetre, fid the rtio of their rdii Sketch the followig grphs : (i) y si x (ii) y t x (iii) y si x π π Prove tht cos + θ cos (π + θ) cot θ + cot(π + θ) 5 θ 9θ 5 θ Prove tht cos θ cos cos θ cos si 5 θ si π x x If t x, π < x <, fid the vlue of si d cos α π cot α 9α 7 prove tht + cos cot α sec cosec α α π cot Prove tht, si x si x + cos x cos x cos x p 9 If t α where α β, α beig cute gle, prove tht; (p cosec β q sec β) q p + q t α + t γ si α + si γ 0 If t β t α t γ, prove tht si β si α si γ Show tht: (i) cot 7 or t ( ) ( + ) (ii) t + + or (iii) si 7 ( ) / ( 5 ) / of 9 TRIGONO METRIC RATIO & IDENTITY TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL

8 Prove tht, t α + t α + t α + cot α cot α If cos (β γ) + cos (γ α) + cos (α β), prove tht cos α + cos β + cos γ 0, si α + si β + si γ 0 si α cos α si α cos α Prove tht from the equlity + follows the reltio + b + b b ( + b) 5 Prove tht: cosec θ + cosec θ + cosec θ + + cosec θ cot (θ/) cot θ Hece or π π π π otherwise prove tht cosec + cosec + cosec + cosec Let A, A,, A be the vertices of sided regulr polygo such tht; + AA AA AA Fid the vlue of 7 If A + B + C π, the prove tht (i) t² A + t² B + t² C (ii) si A si B si C (iii) cos A + cos B + cos C 9 of 9 TRIGONO METRIC RATIO & IDENTITY FREE Dowlod Study Pckge from website: wwwtekoclssescom x by xsiθ bycosθ If + cosθ siθ b, 0 Show tht (x) / + (by) / ( b ) / cos θ si θ 9 If P cos θ + si θ d Q cos θ si θ, the show tht P P si θ cos θ P Q Q si θ cos θ Q d hece show tht P si θ cos θ, Q cos θ si θ 0 If si (θ + α) & si (θ + β) b (0 < α, β, θ < π/) the fid the vlue of cos (α β) b cos(α β) If A + B + C π, prove tht t B t C + t C t A + t A t B + sec A sec B sec C If t α + tα tβ t β + tβ tα, the prove tht ech side is equl to or t α ± t β EXERCISE IV D B A D 5 B C 7 B A 9 B 0 B B B B A 5 C A 7 B D 9 B 0 BC AB BC AC BCD 5 BD BC 7 BC BD EXERCISE V 75 cm r : r : 5 x si 0 x d cos b TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755) , 990 5, BHOPAL