Trigonometric Functions Class 11 Notes Mathematics Chapter 3 - CBSE
Chapter : 3
What Are Trigonometric Functions ?
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Angles
Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of ray after rotation is called the terminal side of the angle. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, the angle is said to be negative.
Degree Measure
If a rotation from the initial side to terminal side is $$\bigg(\frac{1}{360}\bigg)^{\text{th}}$$
of a revolution, the angle is said to have a measure of one degree, written as 1°.
1° = 60' (60 minute)
1' = 60" (60 second)
Radian Measure
Angle subtended at the centre by an arc of length 1 unit in a unit circle is said to have a measure of 1 radian. If in a circle of radius r, an arc of length l subtends an angle θ radians, the l = rθ.
Relation between degree and radian
2π radian = 360°,
π radian = 180°
$$\therefore\space\text{1 radian}=\frac{180\degree}{\pi}\\\text{Also 1\degree}=\frac{\pi}{180\degree}\text{radian}$$
Trigonometric Functions
Consider a unit circle with centre O at origin of the co-ordinate axis. P(a, b) by any point on the circle with angle ∠AOP = θ.
Therefore, sin θ = b, cos θ = a Since ∆OMP is a right triangle,
we have
OM2 + MP2 = OP2
a2 + b2 = 1
cos2 θ + sin2 θ = 1
sin x = 0 implies that x = nπ
where n is any integer.
$$\text{cos x = 0 implies x (2n + 1)}\frac{\pi}{2}\\\text{where n is any integer.}$$
Sign of Trigonometric Function
We can find trigonometric functions in different quadrant. We have the following table:
| I (0° – 90°) | II (91° – 180°) | III (181° – 270°) | IV (271° –360°) | |
| sin x | + | + | – | – |
| cos x | + | – | – | – |
| tan x | + | – | + | – |
| cosec x | + | + | – | – |
| sec x | + | – | – | + |
| cot x | + | – | + | – |
Domain And Range Of Trigonometric Function
From the definition of sine and cosine functions, we observes that they are defined for all real numbers.
We observe that for each real number x, – 1 ≤ sin x ≤ 1 and – 1 ≤ cos x ≤ 1.
Therefore, domain of sin x and cos x is the set of all real numbers and range is the interval [– 1, 1]
domain of cosec x is {x : x ∈ R, x ≠ np and n∈Z}and range is {y : y ∈ R, y ≤ – 1 or y ≥ 1}.
The domain of tan x is the set.
$$\begin{Bmatrix}x: x∈ R \space x≠ (2n+1)\\\frac{\pi}{2}, n\epsilon Z\end{Bmatrix}$$
and range is the set of all real numbers.
Trigonometric Functions Of Sum And Difference Of Two Angles
$$\centerdot\space\text{sin(– x) = – sin x}\\\centerdot\space\text{cos(– x) = – cos x}\\\centerdot\space\text{cos(x + y)}= \\\text{cos x cos y – sin x sin y}\\\centerdot\text{cos(x – y) =}\\\text{cos x cos y + sin x sin y}\\\centerdot\space\text{cos}\bigg(\frac{\pi}{2}-x\bigg)=\text{sin x}\\\centerdot\space\text{sin}\bigg(\frac{\pi}{2}-x\bigg)=\text{cos x}\\\centerdot\space\text{sin(x + y) =}\\\text{ sin x cos y + cos x sin y}\\\centerdot\text{sin(x – y)} =\\\text{sin x cos y – cos x sin y}$$
$$\centerdot\space\text{sin}\bigg(\frac{\pi}{2}+x\bigg)=\text{cos x},\\\text{cos}\bigg(\frac{\pi}{2}+x\bigg)=-\text{sin x}$$
sin(π – x) = sin x, cos(π – x) = – cos x
sin(π + x) = – sin x, cos(π + x) = – cos x
sin(2π – x) = – sin x, cos(2π – x) = cos x
$$\centerdot\space\text{tan}(x+y) =\\\frac{\text{tan x + tan y}}{1 -\text{tan x . tan y}}\\\centerdot\space\text{tan}(x-y)=\\\frac{\text{tan x - tany}}{\text{1 + tan x. tany}}\\\centerdot\space\text{cot}(x +y)=\frac{\text{cot} \space x.\text{cot} \space y-1}{\text{cot y + cot x}}\\\centerdot\space\text{cot(x – y) = }\frac{\text{cot x. cot y} +1}{\text{cot y. cot x}}\\\centerdot\space\text{cos 2x = cos}^{2}x – \text{sin}^{2}x\\=\text{2 cos}^{2}-1\\ = 1 - 2\text{sin}^{2}x =\frac{\text{1 - tan}^{2}x}{\text{1 + tan}^{2}x}\\\centerdot\space\text{sin 2x = 2 sin x cos x =}\\\frac{\text{2 tan x}}{\text{1 + tan}^{2}x}$$
$$\centerdot\space\text{tan 2x} =\frac{\text{2 tan x}}{\text{1 - tan}^{2}x}\\\centerdot\space\text{sin 3x = 3 sin x – 4 sin}^{3}\text{x}\\\centerdot\space\text{cos 3x} = 4\space\text{cos}^{3}x -\text{3 cos x}\\\centerdot\space\text{tan 3x} =\\\frac{\text{3 tan x - tan}^{3}x}{\text{1 - 3\space}\text{tan}^{2}x}\\\centerdot\space\text{(a) cos x + cos y =}\\\text{2 cos}\bigg(\frac{x+y}{2}\bigg)\text{cos}\bigg(\frac{x-y}{2}\bigg)$$
(b) cos x – cos y =
$$\text{-2 sin}\bigg(\frac{x+y}{2}\bigg)\text{cos}\bigg(\frac{x-y}{2}\bigg)$$
$$\text{(c)}\text{sin x + sin y =}\\\text{2 sin}\bigg(\frac{x+y}{2}\bigg)\text{cos}\bigg(\frac{x+y}{2}\bigg)\\\text{(d)}\space\text{sin x – sin y =}\\\text{2 cos}\bigg(\frac{x+y}{2}\bigg)\text{sin}\bigg(\frac{x+y}{2}\bigg)$$
$$\centerdot\space\text{(a)\space2 cos x cos y =}\\\text{cos(x + y) + cos(x – y)}\\\text{(b) –2 sin x sin y =}\\\text{cos}(x+y)-\text{cos}(x-y)\\\text{(c)\space}\text{2 sin x cos y =}\\\text{sin}(x+y)+\text{sin}(x-y)\\\text{(d)\space}\text{2 cos x sin y =}\\\text{sin(x + y) – sin(x − y)}\\\centerdot\space\text{sin x = 0 gives x = nπ}\\\text{where\space n}\epsilon Z\\\centerdot\space\text{cos x = 0 gives x =\\ (2n + 1)}\frac{\pi}{2}\\\space\text{where n}\epsilon Z\\\centerdot\space\text{sin x = sin y implies x = }\\\text{n}\pi +(\normalsize-1)^{n}y\\\text{where n ∈ Z}\\$$
$$\centerdot\space\text{cos x = cos y implies}\\\text{x = 2np ± y where n ∈ Z}\\\centerdot\space\text{tan x = tan y implies}\\\space\text{x = nπ + y where n ∈ Z.}$$
