`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.

`\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.

`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)`

` 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)`

`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

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

`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)`

`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)`