1. 1
Class XI: Maths
Chapter 3: Trigonometric Functions
Top Formulae
180o
1. 1 radian = = 57o16 ' approximately
π
π
2. 1o = radians = 0.01746 radians approximately
180o
3.
s= r θ
Length of arc= radius × angle in radian
This relation can only be used when θ is in radians
π
4. Radian measure= × Degree measure
180
180
5. Degree measure = × Radian measure
π
6. Trigonometric functions in terms of sine and cosine
1
cos ec x = , x ≠ nπ, where n is any int eger
sin x
1 π
s ec x = , x ≠ (2n + 1) , where n is any int eger
cos x 2
sin x π
tan x = , x ≠ (2n + 1) , where n is any int eger
cos x 2
1
cot x = , x ≠ nπ, where n is any int eger
tan x
7. Fundamental Trigonometric Identities
sin2x + cos2x = 1
1 + tan2x = sec2 x
1 + cot2x = cosec2x
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2. 2
8 Values of Trigonometric ratios:
π π π π 3π
0° π 2π
6 4 3 2 2
1 1 3
sin 0 10 0 –1 0
2 2 2
3 1 1
cos 1 0 –1 0 1
2 2 2
1 not not
tan 0 1 3 0 0
3 defined defined
9. Domain and range of various trigonometric functions:
Function Domain Range
 π π
y = sin x − 2 , 2  [–1, 1]
 
y = cos x 0, π
  [–1, 1]
 π π
y = cosec x  − 2 , 2  − {0} R – (–1,1)
 
π
y = sec x 0, π −  
  R – (–1, 1)
2
 π π
y = tan x − 2 , 2  R
 
y = cot x ( 0, π ) R
10. Sign Convention
I II III IV
sin x + + – –
cos x + – – +
tan x + – + –
cosec x + + – –
sec x + – – +
cot x + – + –
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3. 3
11. Behavior of Trigonometric Functions in various Quadrants
I quadrant II quadrant III quadrant IV quadrant
increases from decreases from decreases from increases from
sin
0 to 1 1 to 0 0 to –1 –1 to 0
decreases from decreases from increases from increases from
cos
1 to 0 0 to –1 –1 to 0 0 to 1
increases from increases from increase from 0 increases from
tan
0 to ∞ –∞ to 0 to –∞ –∞ to 0
decrease from decreases from decreases from decreases from
cot
∞ to 0 0 to –∞ ∞ to 0 0 to –∞
increases from increase from decreases from decreases from
sec
1 to ∞ –∞ to –1 –1 to –∞ ∞ to 1
decreases from increases from increases from decreases from
cosec
∞ to 1 1 to ∞ –∞ to –1 –1 to –∞
12. Basic Formulae
(i) cos (x + y) = cos x cos y – sin x sin y
(ii) cos (x - y) = cos x cos y + sin x sin y
(iii) sin (x + y) = sin x cos y + cos x sin y
(iv) sin (x – y) = sin x cos y – cos x sin y
π
If none of the angles x, y and (x + y) is an odd multiple of , then
2
tan x + tan y
(v) tan (x + y) =
1 − tan x tan y
tan x − tan y
(vi) tan (x – y) =
1 + tan x tan y
If none of the angles x, y and (x + y) is a multiple of π, then
cot x cot y − 1
(vii) cot (x + y) =
cot x cot y
cot x cot y − 1
(viii) cot (x – y) =
cot y − cot x
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4. 4
13. Allied Angle Relations
π 
cos  − x  = sin x
2 
π 
sin  − x  = cos x
 2 
π  π 
con  + x  = –sin x sin  + x  = cos x
2  2 
cos (π – x) = –cos x sin (π – x) = sin x
cos (π + x) = –cos x sin (π + x) = –sin x
cos (2π – x) = cos x sin (2π – x) = –sin x
cos (2nπ + x) = cos x sin (2nπ + x) = sin x
14. Sum and Difference Formulae
x+y x−y
(i) cos x + cos y = 2 cos cos
2 2
x+y x−y
(ii) cos x – cos y = −2 sin 2 sin 2
x+y x−y
(iii) sin x + sin y = 2 sin cos
2 2
x+y x−y
(iv) sin x – sin y = 2 cos 2 sin 2
(v) 2cos x cos y = cos (x + y) + cos (x – y)
(vi) –2sin x sin y = cos (x + y) – cos (x – y)
(vii) 2sin x cos y = sin (x +y) + sin (x – y)
(viii) 2cos x sin y = sin (x + y) – sin (x – y)
15. Multiple Angle Formulae
1 − tan2 x
(i) cos 2x = cos2 x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x =
1 + tan2 x
2 tan x
(ii) sin 2x = 2 sin x cos x =
1 + tan2 x
2 tan x
(iii) tan 2x =
1 − tan2 x
(iv) sin 3x = 3 sin x – 4 sin³ x
(v) cos 3x = 4 cos³ x – 3 cos x
3 tan x − tan3 x
(vi) tan 3x =
1 − 3 tan2 x
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5. 5
16. Trigonometric Equations
No. Equations General Solution Principal Value
1 sin θ = 0 θ = nπ, n∈Z θ=0
cos θ = 0 π π
θ = (2n + 1) , θ=
2 2 2
n∈Z
3 tan θ = 0 θ = nπ θ=0
sin θ = sin α θ = nπ + (-1)ⁿ α θ=α
4
n∈Z
5 cos θ = cos α θ = 2nπ ± α n∈Z θ = 2α, α > 0
6 tan θ = tan α θ = nπ + α n∈Z θ=α
14. (i) sin θ = k = sin (nπ + (–1)ⁿ α), n є Z
θ = nπ + (–1)ⁿ α, n є Z
cosec θ = cosec α ⇒ sin θ = sin α
θ = nπ + (–1)ⁿ α, n є Z
(ii) cos θ = k = cos (2nπ ± α), n є Z
θ = 2nπ ± α, n є Z
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