• Angle 

  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 the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.

      
                      Figure 1(i) - Positive Angle            Figure 1(ii) Negative Angle   

• Degree measure: If a rotation from the initial side to terminal side is (1360)th of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″.Thus, 1° = 60′,1′= 60″
• Radian measure: There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) 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 radian at the centre, we have l=θ/r.
• Relation between degrees and radians : 2π radian = 360°or  π radians = 180°

• Trigonometric ratios

A right angled triangle

• sinθ= perpendicular/hypotenuse

• cosθ= base/hypotenuse

• tanθ= perpendicular/base , θ ≠  (π/2)(2n +1) where n is any integer

• cotθ=1/tanθ , θ ≠nπ, where n is any integer

• secθ=1/cosθ , θ ≠ (π/2)(2n + 1) where n is any integer

• cosecθ=1/sinθ , θ ≠ nπ where n is any integer.

• Signs of trigonometric functions:

• Domains and ranges of trigonometric functions:

• Domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1, 1], i.e., – 1 ≤ y ≤ 1.
• The domain of y = cosec x is the set {x: x ϵ R and x ≠ n π, n ϵ Z} and range is the set {y: y ϵ R, y ≥ 1 or y ≤– 1}.
• The domain of y = sec x is the set {x: x ϵ R and x ≠ π2(2n + 1), n ϵ Z} and range is the set{y: y ϵ R, y≤ -1or y ≥ 1}.
• The domain of y = tan x is the set {x: x ϵ R and x ≠  π2(2n +1), n ϵ Z} and range is the set of all real numbers.
• The domain of y = cot x is the set {x: x ϵ R and x ≠n π , n ϵ Z} and the range is the set of all real numbers.

• Monotonicity of trigonometric functions:

• Plots of trigonometric functions:

• y=sinx

• Basic Identities:

• sin2 x + cos2 x = 1
• 1 + tan2 x = sec2 x
• 1 + cot2 x = cosec2 x
• cos (– x) = cos x
• sin (– x) = – sin x

• Trigonometric Functions of Sum and Difference of Two Angles

• cos (x + y) = cos x cos y – sin x sin y
• cos (x – y) = cos x cos y + sin x sin y
• cos (π2  – x) = sin x & sin (π2  – x) = cos x
• sin (x + y) = sin x cos y + cos x sin y
• sin (x – y) = sin x cos y – cos x sin y
• cos (π2  + x) = – sin x  & sin (π2  + x) = cos x
• 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
• tan (x + y) = tanx+tany1-tanxtany
• tan (x – y) =tanx-tany1+tanxtany
• cot ( x + y) =cotxcoty-1coty+cotx
• cot (x – y)= cotxcoty+1coty-cotx

• Double Angle Formulae

• cos 2x = cos2x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x = 1-tan2x1+tan2x
• sin 2x = 2 sinx cos x =2tanx1+tan2x
• tan 2x =2tanx1-tan2x

• Triple Angle Formulae

• sin 3x = 3 sin x – 4 sin3 x
• cos 3x= 4 cos3 x – 3 cos x
• tan 3x =3tanx-tan3x1-3tan2x

• Sum to Product Formulae

• cos x + cos y =2cos(x + y2) cos(x- y2)
• cos x – cos y = –2sin(x + y2) sin(x- y2)
• sin x + sin y = 2sin(x + y2) cos(x- y2)
• sin x – sin y =2cos(x + y2) sin(x- y2)

• Product to Sum Formulae

• 2 cos x cos y = cos (x + y) + cos (x – y)
• –2 sin x sin y = cos (x + y) – cos (x – y)
• 2 sin x cos y = sin (x + y) + sin (x – y)
• 2 cos x sin y = sin (x + y) – sin (x – y).

• General solutions of trigonometric equations

• For any real numbers x and y, sin x = sin y implies x = nπ + (–1)n y, where n ϵ Z
• For any real numbers x and y, cos x = cos y, implies x = 2nπ ± y, where nϵZ
• if x and y are not odd multiple of  π2, then tan x = tan y implies x = nπ + y, where n ϵ Z

• Principle solutions

Those general solutions which lie between 0 to 2π i.e. 0≤x≤2 π.