Trigonometric Equation 2.
13
SECTION - A π
GENERAL SOLUTION 7. If x 0, , the number of solutions of the
2
1. The general solution of the equation, equation, sin 7x + sin 4x + sin x = 0 is
2cos2x = 3.2cos2x – 4 is (A) 3 (B) 5
(A) x = 2n, n (B) x = n, n
(C) 6 (D) None of these
(C) x = n/4, n (D) x = n/2, n
2. The solution set of the equation cos3θ 1
8. = if
2cos2θ - 1 2
4sin . cos – 2 cos – 2 3 sin + 3 = 0 in the
interval (0, 2) is
π π
(A) = n + , n (B) = 2n , n
3 3
3π 7π π 5π
(A) , (B) , π π
4 4 3 3 (C) = 2n ± , n (D) = n + , n
6 6
3π π 5π π 5π 11π
(C) , π, , (D) , ,
4 3 3 6 6 6 9. sin 3 = 4 sin . sin 2 . sin 4 in 0 has
(A) 2 real solutions (B) 4 real solutions
(C) 6 real solutions (D) 8 real solutions
3. Total number of solutions of sin x . tan 4x = cos x
belonging to (0, ) are
(A) 4 (B) 7 tan3x - tan2x
10. The set of values of x for which =1
(C) 8 (D) 5 1 + tan3xtan2x
is
4. All solutions of the equation, 2 sin + tan = 0 are (A)
obtained by taking all integral values of m and n in (B) (/4)
2π (C) {n + /4 | n = 1, 2, 3.......}
(A) 2n + , n (D) {2n + /4 | n = 1, 2, 3 .......}
3
2π 11. The number of solutions of
(B) n or 2m ± where n, m
3 sin + 2sin 2 + 3sin 3 + 4sin 4 = 10 in (0, ) is
π (A) 1 (B) 2
(C) n or m ± where n, m (C) 4 (D) 0
3
π SECTION - B
(D) n or 2m ± where n, m TRIGONOMETRIC GRAPH EQUATION
3
5. The most general solution of tan = – 1 and 12. The number of solutions of
|cos x| = sin x, 0 x 4 is -
1 (A) 8 (B) 4
cos = is
2 (C) 2 (D) None
7π 7π
(A) n + , n (B) n + (–1)n , n 13. No. of solution in the equation x = 4 sinx when x
4 4 [0, 2]-
7π (A) 1 (B) 2
(C) 2 n + , n (D) None of these (C) 3 (D) 4
4
6. If 2 cos2 ( + x) + 3 sin ( + x) vanishes then the 14. Find the number of solution of the equation
values of x lying in the interval from 0 to 2 are 30 |sin x| = x in 0 x 2
(A) y = /6 or 5/6 (B) x = /3 or 5/3 (A) 4 (B) 2
(C) x = /4 or 5/4 (D) x = /2 or 5/2 (C) 8 (D) 6
www.motioniitjee.com
Trigonometric Equation 2.14
15. Total number of solution of the equation 20. The number of all possible triplets (a1, a2, a3) such
that : a1 + a2 cos 2x + a3 sin2x = 0 for all x is
5
3x + 2 tan x = in x [0, 2] = (A) 0 (B) 1
2
(C) 2 (D) infinite
(A) 1 (B) 2
(C) 3 (D) 4
21. The value ‘a’ for which the equation
4cosec2 ((a + x)) + a2 – 4a = 0 has a real solution is :
SECTION - C
(A) a = 1 (B) a = 2
TRIGONOMETRIC INEQUALITIES
(C) a = 3 (D) None of these
1
16. The set of solution satisfying inequality |sin x| < is-
2 22. The arithmetic mean of the roots of the equation
4cos3x – 4cos2x – cos( + x) – 1 = 0 in the interval
(A) n, n 6 (n I) (B) 2n, 2n 6 [0, 315] is equal to
(A) 49 (B) 50
(C) 51 (D) 100
5
(C) n 6 , n 6 (D) None of these 23. The solution of |cos x| = cosx – 2sinx is
π
(A) x = n, n (B) x = n+ , n
4
17. The set of solution satisfying inequality π π
2cos2+ sin 2, where /2 (C) x=n+ (–1)n , n (D) (2n+1)+ , n
4 4
C
DNone of these
The set of solution satisfying inequality
sin 3cosθ 1, -π < θ π
3 , 6 ,
C , DNone of these
6
SECTION - D
MIXED PROBLEMS
1
19. A triangle ABC is such that sin(2A + B) = .
2
If A, B, C are in A.P. then the angle A, B, C are
respectively
5π π π π π 5π
(A) , , (B) , ,
12 4 3 4 3 12
π π 5π π 5π π
(C) , , (D) , ,
3 4 12 3 12 4
www.motioniitjee.com
Trigonometric Equation 2.15
GENERAL SOLUTION/ 8. If 2 tan2x – 5 sec x – 1 = 0 has 7 different roots in
TRIGONOMETRIC GRAPH EQUATION
nπ
0, 2 , n N, then greatest value of n is
1. The general solution of the equation
π 2π (A) 8 (B) 10
tan x + tan x + + tan x + = 3 is (C) 13 (D) 15
3 3
nπ π nπ π 7π
(A) + , n (B) + , n 9. If 20 sin2 + 21 cos – 24 = 0 & < < 2 then
4 12 3 6 4
nπ π θ
(C) + , n (D) None of these the values of cot is
3 12 2
15
2. The number of values of x in [0, 5] satisfying
(A) 3 (B)
3
the equation 3cos 2x – 10cos x + 7 = 0 are-
(A) 5 (B) 6 15
(C) 8 (D) 10 (C) – (D) – 3
3
3. Number of solutions of the equation
tan x + sec x = 2 cos x in (– , 2), is
10. The values of x between 0 and 2 which satisfy the
(A) 1 (B) 2 (C) 3 (D) 4 equation sinx . 8 co s 2 x = 1 are in A.P. The
common difference of the A.P. is
4. The number of solution of the equation π π
(A) (B)
|sin x| = |cos 3x| in [–2, 2] is 8 4
(A) 32 (B) 28 3π 3π
(C) 24 (D) 30 (C) (D)
8 4
5. Number of real solutions to the equation sin
(6x) = x, is
(A) 13 (B) 11
(C) 9 (D) 7
TRIGONOMETRIC INEQUALITIES/
MIXED PROBLEMS
6. The set of values of x for which
sin x . cos3x > cos x . sin3x in [0, 2], is
(A) (0, ) (B) 0 ,
4
(C) , (D) none of these
4
7. The number of integral values of a for which the
equation cos 2x+a sin x = 2a – 7 possesses a solution
is
(A) 2 (B) 3
(C) 4 (D) 5
www.motioniitjee.com
Trigonometric Equation 2.16
GENERAL SOLUTION MIXED PROBLEMS
1. sin x + sin 2x + sin 3x = 0 if 4. sin x, sin 2 x, sin 3x are in A.P. if
(A) sin x = 1/2 (B) sin 2x = 0 (A) x = n/2, n (B) x = n, n
(C) sin 3x = (C) x = 2n, n (D) x = (2n+1), n
3 /2 (D) cos x = – 1/2
5. sin x – cos2 x – 1 assumes the least value for the
2. The equation set of values of x given by
x x (A) x = n + (–1)n + 1 (/6), n
2 sin . cos2x + sin2x = 2 sin . sin2x + cos2x (B) x = n + (–1)n (/6), n
2 2
has a root for which (C) x = n + (–1)n (/3), n
(A) sin 2x = 1 (B) sin 2x = –1 (D) x = n – (–1)n (/6), n
1 1 6. If sin(x – y) = cos (x + y) = 1/2 then the values
(C) cos x = (D) cos 2x = –
2 2 of x & y lying between 0 and are given by
(A) x = /4, y = 3/4 (B) x = /4, y = /12
3. sin2 x + 2 sin x cos x – 3 cos2x = 0 if (C) x = /4, y = 5/12 (D) x = 11/12, y = 3/4
(A) tan x = 3 (B) tan x = –1
(C) x = n + /4, n (D) x = n + tan–1 (–3), n
www.motioniitjee.com
Trigonometric Equation 2.17
SUBJECTIVE
Subjective Type 15. Find the general solution of the equation,
1. What are the most general values of which satisfy
x x
the equations, 2 + tan x . cot + cot x . tan =0
2 2
1
(a) sin =
2 16. Find the principal solution of the trigonometric
equation
(b) tan (x – 1) = 3
1
(c) tan = –1 cot3x + sin 2 x - + 3co sx + sin x - 2
4
2
(d) cosec = 3x 2
3 = sin -
2 2
(e) 2cot2 = cosec2
2. Solve : sin 9 = sin 17. Solve the equation : 2 sin x = 3 x2 + 2 x + 3.
3. Solve : cot + tan = 2 cosec 18. Solve the inequality sin 2x> 2 sin2x+(2– 2 ) cos2x.
4. Solve : sin2 = cos 3
19. Find the general solution of the equation,
5. Solve : cot = tan 8
tan2(x + y) + cot2(x + y) = 1 – 2x – x2.
6. Find all the angles between 0º and 90º which satisfy
20. Solve the equation for x,
the equation sec2 . cosec2 + 2 cosec2 = 8
1 1 1
+ log 5 (sinx) + log15 cosx
52 + 52 = 15 2
7. Solve : sin + sin 3 + sin 5 = 0
21. Determine the smallest positive value of x which
8. Solve : cos2x + cos2 2x + cos2 3x = 1.
satisfy the equation, 1 sin 2 x 2 cos 3 x 0 .
9. Solve : sin2n – sin2(n – 1) = sin2, where n is
constant and n 0, 1 22. Given that A, B are positive acute angle, solve : 3
10. Solve : tan + tan 2 + 3 tan tan 2 = 3 2 2 3 -1
sin 2 A = sin 2B & 3 sin A + sin B = .
2
11. Find all value of , between 0 & , which satisfy 23. Find the set of values of 'a' for which the equation,
the equation; cos . cos 2 . cos 3 = 1/4 sin4 x + cos4 x + sin 2x + a = 0 possesses solutions.
Also find the general solution for these values of
'a'.
12. Solve for x, the equation 13 - 18tanx =6 tan x–3,
where –2 < x < 2. 24. Match the following for number of solutions in [0, 2]
Column - I Column - II
13. Find all the solutions of, (A) sin2 – tan2 = 1 (P) 2
4 cos² x sin x 2 sin² x = 3 sin x. (B) sin + cos = 1 (Q) 0
(C) tan + sec = 2cos (R) 3
14. Solve the equation for 2
(D) 3sin – 4 sin + 1 = 0 (S) 1
2
0 2; (sin 2 + 3 cos 2) – 5
π
= cos 6 - 2θ
www.motioniitjee.com
Trigonometric Equation 2.18
1. The possible vslues of (0, ) such that sin ()
+ sin (4) + sin (7) = 0 are [AIEEE 2011] 2. If 0 x < 2, then the number of real values of x,
which satisfy the equation
2π π 4π π 3π 8π cosx + cos2x + cos3x + cos4x = 0, is :
(A) , , ; , , [MAIN 2016]
9 4 9 2 4 9
(A) 5 (B) 7
(C) 9 (D) 3
π 5π π 2π 3π 8π
(B) , , , , ,
4 12 2 3 4 9
2π π π 2π 3π 35π
(C) , , , , ,
9 4 2 3 4 36
2π π π 2π 3π 8π
(D) , , , , ,
9 4 2 3 4 9
www.motioniitjee.com
Trigonometric Equation 2.19
1. The number of solutions of the pair of equations Let P = { : sin – cos =
2 2
5. 2 cos } and
2sin – cos 2 = 0 and 2cos – 3 sin = 0
in the interval [0, 2] is [JEE 2007, 3] Q = { : sin + cos = 2 sin } be two sets.
(A) zero (B) one Then
(C) two (D) four [JEE 2011]
(A) P Q and Q – P (B) Q P
2. For 0 < < /2, then solution(s) of (C) P Q (D) P = Q
6
cosec ( + (m – 1)/4) cosec( + m/4) = 4 2 6. For x (0, ), the equation sinx + 2sin 2x – sin 3x = 3 has
m =1 [JEE 2014]
is(are) [JEE 2009] (A) infinitely many solutions (B) three solutions
(A) /4 (B) /6 (C) one solution (D) no solution
(C) /12 (D) 5/12
7 The number of distinct solutions of equation
3. The number of values of in the interval 5
cos22x + cos4x + sin4x + cos6x + sin6x = 2
π π 4
nπ
- 2 , 2 such that θ for n=0, ±1, ±2 and in the interval [0, 2] is [JEE 2015]
5
tan = cot 5 as well as sin2 = cos4 is [JEE 2010]
8. Let S = x , : x 0, , The sum of
4. The number of all possible values of when (0) 2
for which the system of equation all distinct solutions of the equation 3 sec
(y + z) cos 3 = (xyz) sin 3 [JEE 2010]
x + cosec x + 2 (tan x – cot x) = 0 in the set
2cos3θ 2sin3θ S is equaL to [JEE 2016]
x sin3 = +
y z 7 2
(A) (B)
(xyz) sin 3 = (y + 2z)cos 3 + y sin 3 have a 9 9
solution (x0, y0, z0) with y0,z0 0 is 5
(C) 0 (D)
9
www.motioniitjee.com
Trigonometric Equation 2.20
EXERCISE - I
JEE Main
1 B 2 D 3 D 4 B 5 C 6 A 7 B
8 B 9 D 10 A 11 D 12 B 13 B 14 A
15 C 16 D 17 C 18 B 19 B 20 D 21 B
22 B 23 D
EXERCISE - II
JEE Advance
Single correct Option - type Questions
1 C 2 C 3 B 4 C 5 B 6 B 7 D
8 D 9 D 10 B
Multiple correct Option - type Questions
1 B,D 2 A,B,C,D 3 C,D 4 A,B,C,D 5 A,D 6 B,C,D
EXERCISE - III
Subjective - type Questions
1. (a) n + (–1)n , n (b) n + , n (c) n – , n
4 3 4
(d) n + (–1)n , n n ± , n (e) n ± , n
3 4 4
m (2m 1)
2. , m or , m 3. 2n ± , n
4 10 3
1 1
4. 2n , n or 2n – , n 5. n , n 6. 45º and 60º
2 5 2 2 9
n 1
7. , n or n , n 8. x = (2n + 1) , n or x (2n + 1) , n or x = n ± , n
3 3 4 2 6
m 1 1
9. m, m or , m or m , m 10. n ,n 11.
n 1 2 n 33
3 5 2 7 2
, , , , , 12. , where tan = 13. n ; n + (–1)n
8 3 8 8 3 8 3 10
3 7 19 2
or n + (–1)n 14. = , 15. x = 2n ± , n
10 12 12 3
16. x = /6 17. 18. n + < x < n + 19. x = –1, y = n ± + 1 20.
8 4 4
1
x = 2n + , n only 21. x = /16 22. A = 15º, B = 30º 23. [n
6 2
3 1
+ (–1)n sin–1 (1 – 2a 3 )] where n and a 2 , 2
24. (A) (Q), (B) (R), (C) (P), (D) (R)
EXERCISE - IV
Previous Year’s Question
JEE Main
1 D 2 B
JEE Advanced
1. C 2. C,D 3. 3 4. 3 5. D 6. D 7. 8
8. C
www.motioniitjee.com
