﻿ Class 11 NCERT Math Solution
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TOPICS
Formulae
Basic Formula :-
```   If in a circle of radius r, an arc of length l subtends an angle of θ radians, then
1. Arc of Length (l) = r θ
2. Angle of θ radians (θ) = l/r
3. Circle of radius (r) = l/θ
```
Notational Convention :-
```1. 1 Radian = 180°/π = 57° 16′ approximately
2. 1 Degree = π/180 Radian = 0.01746 Radian approximately
3. Radian Measure = 180°/π x Degree Measure
4. Degree Measure = π/180 x Radian Measure
5. 2π radian = 360°
6. π radian = 180°
7. 1° (Degree) = 60′ (Minutes)
8. 1′ (Minute) = 60″ (Seconds)
```
Trigonometric Functions :-
```1. sin (2nπ + x) = sin x, n ∈ Z
2. cos (2nπ + x) = cos x, n ∈ Z
3. sin x = 0 implies x = nπ, where n is any integer
4. cos x = 0 implies x = (2n + 1)π/2 , where n is any integer.
5. cosec x = 1/sin x , x ≠ nπ, where n is any integer.
6. sec x = 1/cos x , x ≠ (2n + 1)π/2, where n is any integer.
7. tan x = sin x/cos x, x ≠ (2n +1)π/2, where n is any integer
8. cot x = cos x/sin x, x ≠ n π, where n is any integer.
```
Identities :-
```1. cos (x + y) = cos x cos y – sin x sin y
2. cos (x – y) = cos x cos y + sin x sin y
3. sin (x + y) = sin x cos y + cos x sin y
4. sin (x – y) = sin x cos y – cos x sin y
5. tan (x + y) = (tan x + tan y)/(1 - tan x.tan y)
6. tan (x - y) = (tan x - tan y)/(1 + tan x.tan y)
7. cot (x + y) = (cot x.cot y - 1)/(cot y + cot x)
8. cot (x - y) = (cot x.cot y + 1)/(cot y - cot x)
```
Identities :-
```1. cos x + cos y = 2 cos(x + y)/2 . cos(x - y)/2
2. cos x - cos y = -2 sin(x + y)/2 . sin(x - y)/2
3. sin x + sin y = 2 sin(x + y)/2 . cos(x - y)/2
4. sin x - sin y = 2 cos(x + y)/2 . sin(x - y)/2
```
Identities :-
```1. cos(π/2 - x) = sin x
2. cos(π/2 + x) = -sin x
3. sin(π/2 - x) = cos x
4. sin(π/2 + x) = cos x
5. cos (π – x) = – cos x
6. cos (π + x) = – cos x
7. sin (π – x) = sin x
8. sin (π + x) = – sin x
9. cos (2π – x) = cos x
10. sin (2π – x) = – sin x
```
Identities :-
```1. sin 2x = 2 sin x.cos x
2. sin 2x = 2tan x/(1 + tan² x)
3. cos 2x = cos² x – sin² x
4. cos 2x = 2cos² x – 1
5. cos 2x = 1 – 2 sin² x
6. cos 2x = (1 - tan² x)/(1 + tan² x)
7. tan 2x = 2tan x/(1 - tan² x)
8. sin 3x = 3sin x – 4sin³ x
9. cos 3x = 4cos³ x – 3cos x
10. tan 3x = (3tan x - tan³ x)/(1 - 3tan² x)
```
Identities :-
```1. 2cos x cos y = cos ( x+ y) + cos ( x – y)
2. –2sin x sin y = cos (x + y) – cos (x – y)
3. 2sin x cos y = sin (x + y) + sin (x – y)
4. 2 cos x sin y = sin (x + y) – sin (x – y)
```
Identities :-
```1. sin x = 0 gives x = nπ, where n ∈ Z.
2. cos x = 0 gives x = (2n + 1)π/2 , where n ∈ Z.
3. sin x = sin y implies x = nπ + (– 1)ⁿ y, where n ∈ Z.
4. cos x = cos y, implies x = 2nπ ± y, where n ∈ Z.
5. tan x = tan y implies x = nπ + y, where n ∈ Z.
```
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