KEY CONCEPTS

BASIC TRIGONOMETRIC IDENTITIES :

2 2
(a) sin 9 + c o s 9 = 1 ; -l<sin9<l; -l<cos9<l V 9eR
(b) sec2 9 - tan2 9 = 1 ; | sec 0 | > 1 V 9 e R
(c) cosec2 9 - cot2 9 = 1 ; |cosec9|>l V 9 e R
IMPORTANT T' RATIOS :

(a) sinn7t = 0 ; cosn7T = (-l) n ; tann7t = 0 where n e I

;
(b) . (2n +
sin- l)7i
(-l) n & cos (2n + 1)Tt = Q where n e I

V3-1 __ 5n
(c) sin 15° or sin —r=- = cos 75 0 or cos —
12 2V2 12
V3+1 . _.0 . 5 7X
cos 15° or cos — —r=~ = sm 75° or sin —
12 2V2 12

tan 15° = — - = 2-V3 = cot 75° ; tan75°= = 2+V3 =cot 15°

V3 + 1 V3 - 1

. 11 vJ2 - ^ 71 1/2 + V 2 71 r- 3tt rz ,

(d) sm = • cos — = J L - — - ; tan - = V2 - 1 ;
— tan — = V2 + 1
8 8

• 100 _ V
18 = 5 - 1
71
•
(e) sm — or sin &
P
cos ICO
36 or cos —
71
= V J + l
10 4 5 4

TRIGONOMETRIC FUNCTIONS O F ALLIED ANGLES :

If 9 is any angle, then - 9, 90 ±9 , 180 ±9 , 270 ±9 , 360 ±9 etc. are called ALLIED ANGLES

(a) sin ( - 9) = - sin 9 ; cos ( - 9) = cos 9

sine & c o s e c
(b) sin (90°- 9) = cos 9 ; cos (90° - 9) = sin 9 only + v e All + v e

(c) sin (90°+ 9) = cos 9 ; cos (90°+ 9) = - sin 9

(d) sin(180°-9) = sin9 ; cos (180°-9) = - c o s 9
(e) sin (180°+ 9) = - sin9 ;cos(180°+9) = - c o s 9 tan & cot cos & sec
only + v e only + v e
(f) sin (270°- 9) = - cos 9 ; cos (270°- 9) = - sin 9
(g) sin (270°+ 9) = - cos 9 ; cos (270°+9) = sin9
4. TRIGONOMETRIC FUNCTIONS O F SUM O R DIFFERENCE O F T w o ANGLES

(a) sin (A ± B) = sinA cosB ± cosA sinB

(b) cos (A ± B) = cosA cosB + sinA sinB
(c) sin2A - sin2B = cos2B - cos2A = sin (A+B). sin (A- B)
(d) cos2A - sin2B = cos2B - sin2A = cos (A+B). cos (A - B)
tanA ± tanB cotA cotB + 1
(e) tan (A ± B) : (f) cot (A ± B) =
1 + t a n A tanB " ~ ' cotB + cotA

5. FACTORISATION O F T H E S U M O R DIFFERENCE O F T w o SINES O R COSINES :

C + D C D
(a) sinC + sinD = 2sin C + P
cos——— (b) sinC - sinD = 2 cos sin—
2

D C + D C - D
(c) cosC + cosD = 2 cos C + D
cos (d) cosC-cosD = - 2 s i n sm

^Bansal Classes Trig.-<t>-1 or , ^ ^ Sin [2]

 TRANSFORMATION O F PRODUCTS INTO SUM O R DIFFERENCE O F SINES & COSINES:
(a) 2 sinA cosB = sin(A+B) + sin(A-B) (b) 2 cosA sinB = sin(A+B) - sin(A-B)
(c) 2 cosA cosB = cos(A+B) + cos(A-B) (d) 2 sinA sinB = cos(A-B) - cos(A+B)
MULTIPLE ANGLES A N D H A L F ANGLES :

9 9
(a) sin 2A = 2 sinA cosA ; sin9 = 2 s i n - c o s -
(b) cos 2 A = cos2A - sin2A = 2cos2A - 1 = 1 - 2 sin2A;

cos 6 = cos2— - sin2-^ = 2cos2^- - 1 = 1 - 2sin2^-.

2 2 2 2

. l-cos2A
2 cos2A = 1 + cos 2A, 2sin2A = 1 - cos 2A ; tan2A =
l + cos2A
9 9
2 cos2— = 1 + cos 9 , 2 sin2— = 1 - cos9.
2 2
2? t T
an A ^ tanf
(c)
w
tan2A= ?2 ; tan9= , ~ri
1 - tan A 1 - tan T

2tanA
(d)
W
sin2A = , cos2A= 1 - t a a [ A (e) sin3A = 3 sinA-4 sin3A
1 + tan A 1 + tan A

3 tan A - tan 3 A
(f)
w
cos 3A = 4 cos3A - 3 cosA (g) tan 3A , „ ,
1 - 3 tan'A

8. THREE ANGLES :
tanA + tanB + tanC - tanA tanB tanC
(a) tan (A+B+C) =
1 - tanA tanB - tanB tanC - tanC tanA
N O T E IF : (i) A+B+C = n then tanA+tanB + tanC = tanA tanB tanC
(ii) A+B+C = ~ then tanA tanB + tanB tanC + tanC tanA= 1
(b) If A + B + C = 7t then: (i) sin2A + sin2B + sin2C = 4 sinA sinB sinC
(ii)
ABC
sinA + sinB + sinC = 4 cos — cos — cos —
v
' 2 2 2

9. MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS:

2 2 2 2
(a) Min. value of a tan 9 + b cot 9 = 2ab where 9 € R
(b) Max. and Min. value of acos9 + bsin9 are ,j a 2 52 and—^Ja 2 + ^2
(c) If f(9) = acos(a + 9) + bcos(B + 9) where a, b, a and P are known quantities then
-•\Ja2 + b2 + 2abcos(ct-p) <f(9)< ^/a2 +b 2 + 2abcos(a - (3)

(d) If(x,B e [0,-7 j and a + p = a (constant) then the maximum values of the expression
cosa cosp, cosa + cosp, sina + sinP and sina sinp
a
occurs when a = P = —.

(e) If a,p e fn —J
A and a + p = a(constant) then the minimum values of the expression
o
seca + seep, tana + tanp, coseca + cosecp occurs when a = P = —.
(f) IfA, B, C are the angles of a triangle then maximum value of
sinA + sinB + sinC and sinA sinB sinC occurs when A=B = C = 60°
(g) In case a quadratic in sin9 or cos9 is given then the maximum or minimum values can be interpreted
by making a perfect square.

^Bansal Classes Trig.-(f>-I [3]

10. Sum of sines or cosines of n angles,

/ \ si•n^n
Bf- (, n-1
sin a + sin (a + P) + sin (a + 2p ) + + sin ^a + n - i p j = sin I a +-—-p
sin ^
n

s i n• P /
/ — ( n-1
cos a + cos (a + P) + cos(a + 2p ) + + cos la + n-1 pi = 1- cos a + ——
sin j V 2
EXERCISE-I
Q.l Prove that cos2a + cos2 (a + P) - 2cos a cos p cos (a + P) = sin2P
Q.2 Prove that cos 2a = 2 sin2P + 4cos (a + P) sin a sin p + cos 2(a + P)
Q.3 Prove that, tan a + 2 tan 2a + 4 tan 4a + 8 cot 8 a - cot a .
Q.4 Prove that : (a) tan 20° . tan 40° . tan 60° . tan 80° = 3

(b) tan9° -tan27° -tan63° + tan81° = 4 . (c) sin 4 ~ + sin 4 ^ + sin 4 ~ + sin 4 ~ = |
16 16 16 16 2

Q.5 Calculate without using trigonometric tables :

2 cos40° - cos20°
(a) cosec 10° - S sec 10° (b) 4 cos 20° - V3 cot 20° (c)
sin20°

sec 5° cos 40° . 27t 4TI 671

(d) 2V2 sin 10° 2sin35 (e) c o s — + c o s — + cos
2 sin 5° 7

(f) tan 10° - tan 50° + tan 70°

Q.6 (a) Showthat 4 sin 17° . sin 43° . sin 77° = sin 51°
(b) Prove that sin2120 + sin221° + sin2390 + sin 2 48°= l+sin 2 9° + sin218° .

cot7
Q.7 Showthat: (a) y or tan 82 y = (V3 + V2) (V2 + 1) or V2+V3+V4+V6

(b) t a n l 4 2 y =2 +V2-V3-V6 .

m +n
Q.8 If mtan(9-30°) = ntan(6+ 120°), show that cos20 2(m - n)

cosu
Q.9 If cos9= —— , prove that, tan— = ± tan— .
1-ecosu 2 V1 _ e 2

4 5 7
1
Q.10 If cos (a + P) = j ; sin (a - P) = — & a , P lie between 0 & — , then find the value of tan
2a .
Q.ll Prove that if the angles a & P satisfy the relation p) = ~ (M > lnl) then
. tang

1 + tana _ 1 - tana tanP

m+n m-n
Q.12 (a) If y = 10 cos2x - 6 sin x cos x + 2 sin2x, then find the greatest & least value of y.
(b) If y = 1 + 2 sin x + 3 cos2 x, find the maximum & minimum values of y V x e R.
(c) If y = 9 sec2x +16 cosec2x, find the minimum value of y V x 6 R.

y'vfl/ < a ses Trig.-(f>-1 [4]

Q.13 (a) Prove that 3 cos (e + y j + 5 cos 0 + 3 lies from - 4 & 10 .

(b) Prove that (2V3 + 4) sin 0 + 4 cos 0 lies between - 2(2 + Vs ) & 2(2 + VJ).
Q.14 If a + p = c where a, p > 0 each lying between 0 and 7i/2 and c is a constant, find the maximum or
minimum value of
(a) sin a + sin p (b) sin a sin p
(c) tan a + tan p (d) cosec a + cosec p

Q.15 Let A,., A 2 , , An be the vertices of an n-sided regular polygon such that ;
1
+ —-— . Find the value of n.
A.j A2 ^•i .A] A.^

Q. 16 Prove that: cosec 0 + cosec 20 + cosec 22 0 + + cosec 2 n " 1 0 = cot (0/2) - cot 2 n " ! 0

Q.17 For all values of a , p , y prove that ;

cos a + cos p + cos y + cos (a + p + y)x = 4, cos a+B . cos——

P+Y . cos Y+a .
2 .: 2 .2
18 Sh th t *+ + C0S
^ - 2sinA-2sinB
Q.18 Showthat C0Sj^ 1 - sinB sin(A-B) + cosA - cosB

A i n , _ t a n a + tan y . . s i n 2 a + sin2y
Q.19 IIf( tan QP = — , prove that sin 2B = ; 7-^ •
1 + tana.tany 1 + s i n 2 a . sm2y

Q.20 If a + p = y , prove that cos2 a + cos2 p + cos2 y = 1 + 2 cos a cos P cos y .

( l - t a n f ) ( l - t a n f ) |i l - t a n | j ' _ s i n a + sin(3 + siny — 1
Q.21 If a + p + y = | , show that
(l + t a n f ) (l + tan f)ll 1 + t a n i c o s a + cosp + cosy

Q.22 If A + B + C = 7t and cot0 = cot A + cot B + cot C, showthat,

sin (A - 0). sin (B - 0). sin (C - 0) = sin30 .

% 3n 571 17TC
Q.23
V IfP= cos— + cos— + cos— + + cos and
19 19 19 19
271 47t 6tz 20n
Q= cos—+ cos—+ C
OS—+ + cos-^-? then find P - Q.

- *
Q.24 Without using the surd value for sin 18° or cos 36°, prove that 4 sin 36° cos 18° — V5

Q.25 For any three angles a , p and y prove that:

' cos(a + P) cos(a + y)V ( sin(a + P) sin(a + y)^ 2

cos(a - P) cos(a - y )J \ cos(a - P) cos(a - y)>
= sec2(a - p) • sin2(P - y) • sec2(y - a)

Bamal Classes Trig.-</>-! [5]

 EXERCISE-II
Q.l If tana = p/q where a = 6p , a being an acute angle, prove that ;
- (p cosec 2 P - q sec 2 P) = ^/p2 + q2 .

Q.2 If +— = +— = 1, where 9 & 5 do not differ by an even multiple of n, then prove

cosa sina cosa sina

cosB . cos5 sin9 . sin5

that i + —— +1=0.
cos a sm a
cos 30 + cos3(j)
Q.3 Prove that 2 c o s ( 0 - <j)) - 1 = (cosO + cos<j)) cos(9 + <j>) - (sin9 + sin<()) sin(9 + <j>)

^ ^
Q.4 TX- tan
If * — fn + j-
y) = +tani3 (n— +A- * •siny = •sinxI 3 -—„
prove^that
+ sin2 x
.,
v
U 2J v4 2/ 1 + 3 sm x

Q.5 Show that, J ™ L + + = I (tan27x -tanx)

cos3x cos9x cos27x 2

Q.6 Prove that; cosec x. cosec 2x. sin 4x. cos 6x . cosec 1 Ox =
cos3x cos5x cos7x cos9x
sin2xsin4x
+ s— r:—r~:
in4xsin6x sin6x sin8x
+ sin8xsinl0x

Q.7 If 9= - j , prove that tan9 ,tan29 + tan29 ,tan49 + tan49 ,tan9 = - 7 .

7T CD^Y
Q.8 For 0 < x < - prove that, — r — — > 8.
4 sin x(cosx - sinx)

Q.9 (a) If a = prove that, sin a + sin 2a + sin4a = ~ (b) siny . sin-^- . s i n ^ =

Q.10 For all 9 in show that cos(sin9)>sin(cos9).

3
Q.ll Prove that the value of cos A + cos B + cos C lies between 1 & — where A + B + C = tc.
Q.12 If m2 + m'2 + 2mm'cos9 = l , n2 + n,2 + 2nn'cos9= 1 and
mn + m' n' + (mn' + m' n) cos 9 = 0, then prove that m2 + n2 = cosec2 9.

Q.13 If cos2 9 = m
?
1
& t a n 3 = tan a , prove that cos2/3 a + sin2/3 a = f-^j

Q 14. If V2cosA=cosB + cos3B & sin A = sinB-sin 3 B , showthat sin(A-B) = ± i .

sin a a
Q.15 Prove thatfromthe equality + = — f o l l o w s the relation; s i " 3 a + a
= -—
*

Q.16 Prove that the triangle ABC is equilateral iff, cotA+cotB + cotC= S .
cos a s
Q.17 If ( = ' n ( a3 — = m ^ th e n show that m2 + m cos a = 2 .
cos 9 sin 9
1/2
Q. 18 Prove that: 4 sin 27° = (5 + VJ) - (3 - V?) .

Q.19 If A+B+C = TT ; provethat t a n 2 Y + t a n 2 | + t a n 2 Y >1. [Hint: E t a n - . t a n | = 1]

feBansal Classes Trig.- [6]

Q.20 If A+B+C = n (A, B , C > 0), prove that sin^ • sinf . sin^ < ^ .
2 2 2 8

Q.21 Show that eliminating x&y from the equations, sinx + siny = a ;
8a b
cosx + cosy = b & tanx + tany = c gives =c
4a'

2 s e c 6 + 3 t a n 6 + 5sin 9 - 7 c o s 0 + 5 l-cos0
Q.22 Show that
2 t a n 8 + 3sec 0 + 5 c o s 9 + 7 s i n 0 + 8 sin 0

. x
, tan —
Q.23 Evaluate : £ 2n
n=12n-1COS• 2 n X
-l

f p+ y-a^| rynh a -
Q.24 If a + (3+y = 7t & tan • tan • tan = 1, then prove that;
I 4 J V 4 I 4 j
1 + cos a + cos P + cos y = 0 .

Q.25 V x e R, find the range of the function, f (x) = cos x (sin x + ^sin 2 x + sin2 a ) ; a e [0, n]
EXERCISE-III
Q.l (a) sec29 = 4xy
, is true if and only if :
(X + y ) 2

(A) x+y*0 (B) x = y , x * 0 (C) x = y (D) x * 0 , y * 0

(b) Find all values of 9 in the interval ~ s a t i s f y i n g the equationn;;
(1 - tan 9) (1 + tan 9) sec2 9 + 2tan2 9 = 0 . [JEE'96,1 + 2]

Q.2 (a) Let n be an odd integer. If sin n9 = £ b sinr 9, for every value of 9, then:
r= 0
(A) b0 = 1, b, = 3 (B) bQ = 0, bj = n
(C) b0 = - 1, bj = n (D) b0 = 0, bj = n2 - 3n + 3
(b) Let A0 Aj A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius.
Then the product of the lengths of the line segments A0 Aj, A0 A2 & A0 A4 is :

(B)3V3 (C) 3 (D) ^

(c) Which of the following number(s) is/are rational ?
(A) sin 15° (B) cos 15° (C) sin 15° cos 15° (D) sin 15° cos 75°
(d) Prove that a triangle ABC is equilateral if & only if tan A+tan B + tan C = 3 V3 •
[JEE '98,2 + 2 + 2 + 8 = 14outof200]

Q.3 9 (1+ sec9)(1+ sec29)(1+sec49)....(1 + sec2n9) Then

For a positive integer n, let.fn (9) = ttan—

n f n
(A)/2^]=l (B ) / 3, V32. = 1 WfAm.
1 [JEE '99,3]
Q.4
(a) Let f(9) = sin9 (sin9 + sin 3 9). Then f(9)
(A) > 0 only when 9 > 0 (B) < 0 for all real 9
(C) > 0 for all real 9 (D) < 0 only when 9 < 0.
[ JEE 2000 Screening. 1 out of 35 ]
Bansal Classes Trig.-(j>-1 m
 A B C A B C
(b) Ei any triangle A B C , prove that, c o t — + cot — + cot — = cot — cot — cot — .
Z* Z* Zi Zi Zi

[ JEE 2000 Mains, 3 out of 100 ]

Q.5
(a)
Find real values of x for which, 27 cos2x . 81sul2x is minimum. Also find this minimum value.
(b) Find the smallest positive values of x & y satisfying, x - y = — , cot x + cot y = 2 .
[ REE 2000, 3 + 3 ]
71
Q.6 If a + (3 = — and p + y = a then tana equals
(A) 2(tanP + tany) (B) tanp + tany (C) tanp + 2tany (D) 2tanP + tany
[JEE 2001 (Screening), 1 out of 35 ]
1 1
Q.7 If 9 and (j) are acute angles satisfying sin9 = —, cos <>| = —, then 9 + § e
r
7C 71 f n 2%^ 2n 5nx 571
(A)
3'2 (B) [ 2'T (C)
T'T (D) v6 .
[JEE 2004 (Screening)]
Q. 8(a) In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they A
touch each other and also the sides of the triangle. Area of the triangle is
(A) 4 + 2 V3 (B) 6 + 4 V3

(C) 12 +
7V3 7V3
(D)3 + 4
4 ir
(h) cos(a - P) = 1 and cos(a + P)= 1 /e, where a, P e [ - n, 7t], numbers of pairs of a, P which satisfy both
the equations is
(A)0 (B)l (C) 2 (D)4
[JEE 2005 (Screening)]
ANSWER SHEET
EXERCISE-I
Q 5. (a) 4 (b) -1 (c) V3 (d) 4 (e) (f) fi Q10. | |
13
Q12- (a> ymax= 11
i ymm = 1
(b) ymax= y ; y min
mm = - \ ,' v(c)49
'
2
Q14. (a) max = 2 sin (c/2), (b) max. = sin (c/2), (c) min = 2 tan (c/2), (d) min = 2 cosec (c/2)
Q 15. n = 7 Q23. 1
EXERCISE -II
1
Q 23. Q-25 - V1 + sin2 a ^ y ^ Vl + sin
si a
sin2x n_I X
2 sin n-1

EXERCISE-III
Q.l (a) B (b) ± Q.2 (a) B, (b) C, (c) C Q.3 A, B, C, D Q.4 (a) C
5
Q.5 (a) Minimum value = 3 ~5; maximum value = 3 5 ,(b) x= — ; y = — Q.6 C
7Z 71
Q.7 B
12 6
Q.8 (a) B, (b)D

^Bansal Classes Trig.-1 [8]