Mathematics Definitions , Key Concepts, Points to Remember & Formulas

### Definitions

color {green}{★ ul"Introduction"}

\color{green} ✍️ The word "‘trigonometry’" is derived from the Greek words color{green}{"‘trigon’"} and color{green}{"‘metron’"} and it means color{green}{"‘measuring the sides of a triangle’."}

color {green}{★ ul"Angles "}

\color{green} ✍️ Angle is a measure of rotation of a given ray about its initial point.

\color{green} ✍️ The original ray is called color{blue}{ul"the initial side"} and the final position of the ray after rotation is called color{blue}{ul"the terminal side of the angle .")

\color{green} ✍️ The point of rotation is called color{blue}{ul"the vertex."}

color {green}{★ ul"One Degree "}

If a rotation from the initial side to terminal side is (1/360)^(th) of a revolution, the angle is said to have a measure of color{blue}{ul"one degree"}, written as 1°.

A degree is divided into 60 minutes, and a minute is divided into 60 seconds.

color {green}{★ ul"A Minute "}

● One sixtieth of a degree is called color{blue}{ul"a minute"}, written as 1′.

color {green}{★ ul"A Second "}

● one sixtieth of a minute is called color{blue}{ul"a second"}, written as 1″.

Thus,  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color{green} (1° = 60′, \ \ \ \ \ \ \1′ = 60″)

color {green}{★ ul"1 radian"}

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 color{green}(ul"1 radian").

● We know that the circumference of a circle of radius 1 unit is 2π. Thus, one complete revolution of the initial side subtends an angle of 2π radian.

### Key Concepts

color {green}{★ ul"Reciprocal Identities "}

 color(red)(cosec x = 1/(sin x)) , \ \ color(blue)(x ≠ nπ) , where n is any integer.

 color(red)(sec x = 1/(cos x)) , \ \ color(blue)(x ≠ (2n +1 ) pi/2) , where n is any integer

 color(red)(tan x = (sin x)/(cos x) ), \ \ color(blue)(x ≠ (2n +1) pi/2 ), where n is any integer.

 color(red)(cot x = ( cos x)/(sin x)), \ \ color(blue)(x ≠ n π), where n is any integer

color {green}{★ ul"Pythagorean Identities"}

color(purple)( sin^2 x + cos^2 x = 1)

color(purple)(1 + tan^2 x = sec^2 x)

color(purple)(1 + cot^2 x = cosec^2 x)

### Points to Remember

color {green}{★ ul"θ Radian at the centre"}

if in a circle of radius r, an arc of length l subtends an angle θ radian at the centre,

we have color{green}(θ = l/r) or color{green}(l = r θ.)

color {green}{★ ul"Relation between degree and radian"}

\color{fuchsia} { ul ☛ "Notational Convention"}

 \ \ \ color{blue} {π= 180°} and π/4= 45° are written with the understanding that π and π/4 are radian measures.

Thus, we can say that

color{blue}{"Radian measure" = π/ 180 × "Degree measure"}

color{blue}{"Degree measure "= 180 /π ×" Radian measure"}

color {green}{★ ul"Sign of trigonometric functions "}

Domain and range of trigonometric functions

### Formulas

color {green}{★ ul"Angle-Sum and -Difference Identities "}

sin (x + y) = sin x cos y + cos x sin y

sin (x – y) = sin x cos y – cos x sin y

cos (x + y) = cos x cos y – sin x sin y)

 cos (x – y) = cos x cos y + sin x sin y

tan ( x – y) = (tan x - tan y )/( 1+ tan x tan y)

color{red}(cot (x+ y ) = (cot x cot y -1 )/( cot y + cot x))

cot (x – y)= (cot x cot y + 1)/(cot y - cot x)

color {green}{★ ul"Double-Angle Identities"}

sin 2x = 2 sinx cos x = (2 tan x)/( 1+ tan^2 x)

cos 2x = cos^2 x – sin^2 x = 2 cos^2 x – 1 = 1 – 2 sin^2 x = (1- tan^2 x)/(1+ tan^2 x)

tan 2x = (2 tan x)/(1- tan^2 x)

color {green}{★ ul"Sum Identities "}

(i) color{red}(cos x + cos y = 2 cos( (x+y)/2) cos ((x-y)/2))

(ii) color{red}(cos x – cos y = -2 sin ((x+y)/2) sin ((x-y)/2))

(iii) color{red}(sin x + sin y = 2 sin ((x+y)/2 ) cos ((x-y)/2))

(iv) color{red}(sin x – sin y = 2 cos ((x+y)/2) sin ((x-y)/2))

color {green}{★ ul"Product Identities"}

(i) color{red}(2 cos x cos y = cos (x + y) + cos (x – y))

(ii) color{red}(–2 sin x sin y = cos (x + y) – cos (x – y))

(iii) color{red}(2 sin x cos y = sin (x + y) + sin (x – y))

(iv) color{red}(2 cos x sin y = sin (x + y) – sin (x – y)).

color {green}{★ ul"Multiple Angle Formulas"}

sin 3x = 3 sin x – 4 sin^3 x

cos 3x= 4 cos^3 x – 3 cos x

tan 3x = (3 tan x - tan^3 x)/( 1-3 tan^2 x)