EXERCISE -1
Type-1
1) 4 sin 3x - 1= √5. nπ/3 +(-1)ⁿ π/10
2) sin px + sin qx = 0. 2nπ/(p +q), (2n+1)π/(p - q)
3) 4 sin 4x +1= √5. nπ/4 +(-1)ⁿ π/40
4) 7 sin²x+ 3cos² x = 4. nπ ± π/6
5) Sin 2x - cos 2x= 1. (2n+1)π/2, nπ + π/4
6) Sin 6x = sin 4x - sin 2x. nπ/4, nπ ± π/6
7) Sin 3x + sin 2x + sin x= 0, 0< x < 2π. π/2, 2π/3, π, 4π/3, 3π/2
8) Sin 2x + sin 4x + sin 6x= 0. nπ/4, nπ±, π/3
9) sin x + sin 5x = sin 3x, 0≤ x ≤ π. nπ ± π/6
10) Sin 4x+ Sin 3x + sin 2x + sin x= 0. (2n+1)π/2, (2n+1)π, 2nπ/5
11) Sin 4x = cos 3x + sin 2x= 0, 0< x < π. π/6, π/2, 5π/6
12) Sin 3x + sin 2x + sin x= cosx + cos 2x + cos 3x. nπ/2 + π/8, 2nπ± 2π/3
13) Sin 3x + cos 3x = sin 9x cos 7x = 0. nπ/4, (2n+1)π/24
14) 2Sin 3x sin x = 1. (2n+1)π/4, nπ± π/6
15) Sin 3x cos x= sin 4x. nπ/2, nπ± π/6
16) Sin 3x = sin 5x - 2 sin x cos 2x. nπ/3, nπ
17) 2 sin² x + 3 cos x = 0. 2nπ ± 2π/3
18) 2 sin² x +sin² 2x = 2. (2n+1)π/2, (2n+1)π/4
19) 1+ 2sin x cosx - 2 sin x - cos x = 0, 0≤ x ≤ 2π. 0, π/6, 5π/6, 2π
20) 1 - 2 sin x - 2 cos x + cot x = 0, 0< x <2π. π/6, 3π/4, 5π/6, 7π/4
21) sin 2x tan x + 1 = sin 2x + tan x. nπ + π/4, (4n+ 1)π/4
22) sin x + √3 cos x = √2, 0≤ x ≤2π. 5π/12, 23π/12
23) sin x + cos x = √2, -π < x < π. π/4
24) sin x - √3 cos x = 1. 2nπ+ π/2, 2nπ - 5π/6
25) sin x/sin 2x + cosx /cos 2x= 2. (2n+1)π/6 - a/6, nπ+ a/2
26) Sin x + cos x= √2 cos 2x. 2nπ - π/4, 2nπ/3 +π/12
27) 4 sin x sin 2x sin 3x = sin 4x. nπ/2, (2n+1)π/3
Type-2
1) Cosec x + √3 sec x =4. 2nπ/3 + 5π/18, 2nπ + π/6.
Type -3
2) √2 cos x +1= 0. 2nπ ± 3π/4
3) 2(cos²x - sin²x)= 1. nπ ± π/6
4) 4cos²x + 6sin²x =5. nπ ± π/4
5) cos 4x = sin 3x (4n+1)π/14, (4n-1)π/2
6) cos 3x = sin 2x, 0< x < π. π/10, π/2, 9π/10
7) cos 2x - cos 4x= 0, 0< x < 360. 60, 120, 180, 240, 300
8) Cos 4x = cos 3x - cos 2x. (2n+1)π/6, 2nπ ± π/3
9) cos x + cos 3x + cos 5x + cos 7x = 0. (2n+1)π/8, (2n+1)π/4, (2n+1)π/2
10) cos 9x + cos 7x= cos 5x + cos 3x. (2n+1)π/2, nπ/6, π, nπ/2
11) cos x - sin 3x = cos 2x. 2nπ/3, nπ+ π/4, (4n -1)π/2
12) Cos 6x + cos 4x = sin 3x + sin x. (2n+1)π/2, (4n+1)π/14, (4n-1)π/6
13) Cos 2x - sin 2x = cosx - sin x - 1. nπ + π/4, 2nπ ± π/3
14) Cos 6x + cos 4x + cos 2x +1 = 0, 0< x ≤ π. π/6, π/2, 5π/6, π/4, 3π/4
15) Cosx + sin x = 1/√2. 2nπ + 7π/12, 2nπ - π/12
16) cos x + √3 sin x= √3. 2nπ + π/2, 2nπ +π/6
17) cos x - √3 sin x= 1. 2nπ, 2nπ - 2π/3
18) cos³x - cos x sin x - sin³x= 1. 2nπ, 2nπ - π/2,
19) 4 cos x cos 2x cos 3x = 1/4, 0≤ x ≤π. nπ, nπ/3 ± π/9
20) 2 - cos x = 2 tan (x/2). 2nπ + π/2
21) Cos³x sin 3x + sin³x cos 3x = 3/4. (4n+1)π/8
22) Cos x + sin x = cos 2x + sin 2x. 2nπ/3 + π/6
23) Cos 9x cos 7x = cos 5x cos 3x. nπ/4, nπ/12
Type -4
1) sec x + √2= 0. 2nπ± 3π/4
2) sec 4x - sec 2x = 2. (2n+1)π/2, (2n+1)π/10
3) sec (π/4 + x)+ sec(π/4 - x) = 2√2. 2nπ/3, 2nπ
4) 1+ sec x = cot²(x/2). (2n+1)π, 2nπ ± π/3
5) sec x + 1= (2+ √3) tan x,.0< x < 2π. π/6, π.
Type-5
1) √3 tanx +1= 0. nπ- π/6
2) tan²x = 3cosec²x - 1. nπ ± π/3
3) tan px = cot qx. (2n+1)π/2(p+q)
4) Tan x + cot x= 4. nπ/2 +(-1)ⁿ. π/12
5) Tan x + cot 2x= 2. nπ/2+(-1)ⁿ.π/12
6) tanx - cot x= cosecx. 2nπ ± π/3, (2n+1)π
7) tan²x - (√3+1)tan x +√3 = 0. nπ+ 4, nπ + π/3
8) tan 3x + tan x = 2 tan 2x. nπ/2, nπ
9) tan 3x + tan x + tan 2x= 0. nπ/3, nπ+ k where tan k= 1/√2
10) tan 2x + tan x - tan x tan 2x = 1. nπ/3 + π/12
11) tan(π/4 +x) + tan(π/4- x) = 4. nπ ±π/6
12) (1- tan x)(1+ sin 2x)= (1+ tan x). nπ - π/4, nπ
14) tan x = sin 2x + cos 2x. (2n+1)π/4, nπ - π/4
15) tan²x+ cot²x= 2. nπ ± π/4
Type-6
1) √3 cot x +1= 0. n π - π/3
2) cot x - cot 2x= 2. nπ +(-1)ⁿ. π/12
3) cot x + tan x= 2 sec x, 0< x < 360. 30, 150
4) cot x + cot (π/4 + x) = 2. nπ ± π/6
5) 4 cot 2x = cot² x - tan²x. nπ ± π/4
6) cot(x - a)+ cot(x+ a)= 2 cot x. (2n+1)π/2
EXERCISE -2
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