There are three major types of systems of measurement of angles which are as follows,

1. `"Sexagesimal system :"` (Degree measure) In this system a right angle is divided into 90 equal parts, called degrees. Each degree is divided into 60 equal parts, called minutes and each minute is further divided into 60 equal parts, called seconds.

Thus, 1 right angle= 90 degrees = (90 °), 1 ° = 60 min= (60'), 1' =60s= (6)' ')

2. `"Centesimal system:"` In this system a right angle is divided into 100 equal parts, called grades. Each grade is subdivided into 100 centesimal minutes and each minute is further divided into 100 centesimal seconds.

Thus, 1 right angle = 100 grades =(100 g)

1 grade=100 min=(100')

1 min= 100 s = (100'')

3. `"Circular system :"` In this system the unit of measurement is radian as defined below

(i) One radian, written as 1', is the measure of an angle subtended at the centre of a circle by an arc of length equal.to the radius of the circle.

(ii) The number of radians in an angle subtended by an angle of a circle at the centre is equal to `("arc")/("radius")`.

The relation between the three systems of measurement of an angle is `("Degree")/90 =("Grade")/100 = (2 " Radian")/(pi)`

Thus,

(i) To convert radians into degrees multiply by `(180/pi)`

(ii) To convert degrees into radians multiply by `(pi/180)`

The ratios between different sides of a right angled triangle with respect to its acute angles are called trigonometric ratios.

Let PMR be a right angled triangle at M.

sine `theta = ("Perpendicular")/("Hypotenuse")= q/r` and is written as `sin theta`

cosine `theta= ("Base")/("Hypotenuse") = p/r` and is written as `cos theta`

tangent `theta= ("Perpendicular")/("Base") = q/p` and is written as `tan theta`

cosecant `theta= ("Hypotenuse")/("Perpendicular")= r/q` and is written as `cosec theta`

secant `theta = ("Hypotenuse")/("Base") = r/p ` and is written as `sec theta`

cotangent `theta = ("Base")/("Perpendicular")= p/q` and is written as `cot theta`

(i) `sin theta= 1/(cosec theta)` or `cosec theta= 1/(sin theta)`

(ii) `cos theta= 1/(sec theta)` or `sec theta = 1/(cos theta)`

(iii) `cot theta= 1/(tan theta)` or `tan theta= 1/(cot theta)`

(iv) `tan theta= (sin theta)/(cos theta)`

(v) `cot theta= (cos theta)/(sin theta)`

The following are some fundamental identities

(i) `sin^2 theta+ cos^2 theta=1`

(ii) `1+ tan^2 theta= sec^2 theta`

(iii) `1+ cot^2 theta= cosec^2 theta`

Two angles are said to be allied when their sum or difference is either zero or a multiple of `90°`, i.e. the angles `-theta, 90° ±theta, 180° ±theta, 270° ±theta` and `360° ±theta` are called allied angles.

(i) `sin (A + B) = sin A cos B + cos A sin B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sin (A - B) = sin A cos B - cos A sin B`

(ii) `cos (A + B)= cos A cos B - sin A sin B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ cos (A - B)= cos A cos B + sin A sin B`

(iii) `tan (A + B) =(tan A + tan B)/(1 - tan A tan B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tan (A - B) =(tan A - tan B)/(1 + tan A tan B) `

(iv) `cot (A + B) = (cot A cot B - 1)/( cot B + cot A)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \cot (A - B) = (cot A cot B + 1)/( cot B - cot A)`

(v) `tan(A+B+C) = (tanA+tanB+tanC-tanAtanBtanC)/(1-tanAtanB -tanBtanC-tanCtanA) `

(vi) `sin (A+ B) sin (A-B)= sin^2 A - sin^2 B = cos^2 A - cos^2 A`

(vii) `cos(A+B) cos(A-B) = cos^2 A - sin^2 B = cos^2 B - sin^2 A`

(i) `2sinA cos B =sin (A+ B)+ sin (A -B)`

(ii) `2cos A sin B =sin (A +B)- sin (A -B)`

(iii) `2cosA cosB=cos(A +B)+cos(A -B)`

(iv) `2sinA sinB=cos(A+ B)-cos(A-B)`

(v) `sin C+ sin D = 2 sin ((C+D)/2) cos((C-D)/2)`

(vi) `sin C- sin D = 2 sin ((C-D)/2) cos ((C+D)/2)`

(vii) `cos C+ cos D = 2 cos((C+D)/2) cos((C-D)/2)`

(viii) `cos C- cos D= 2 sin ((C+D)/2) sin ((D-C)/2)`

(i) `sin 2A = 2 sin A cos A =(2 tan A)/(1+ tan^2 A)`

(ii) `cos 2 A = cos^2 A - sin^2 A = 2 cos^2 A-1=1-2 sin^2 A=(1- tan^2 A)/(1+ tan^2 A)`

(iii) `tan 2 A =(2 tan A)/(1- tan^2 A)`

(iv) `cot 2A = (cot^2 A -1)/(2 cot A)`

(v) `sin 3 A = 3 sin A - 4 sin^3 A`

(vi) `cos 3 A = 4 cos^3 A- 3 cos A`

(vii) `tan 3 A =(3 tan A- tan^3 A)/(1- 3 tan^2 A)`

(viii) `cot 3 A =(3 cot A - cot ^3 A)/(1- 3 cot^2 A)`

(i) `sin A = 2 sin \ (A/2) cos \ (A/2) = ( 2 tan \ (A/2))/(1+ tan^2 \ (A/2))`

(ii) `cos A = cos^2 \ (A/2) - sin^2 \ (A/2) = 2 cos^2 \ (A/2)-1= 1- 2 sin^2 \ (A/2) = (1- tan^2 \ (A/2))/(1+ tan^2 \ (A/2))`

(iii) `tan A = ( 2 tan \ (A/2))/(1- tan^2 \ (A/2))`

(iv) `cot A = (cot^2 \ (A/2)-1)/(2 cot \ (A/2))`

(i) `sin 15^o = ( sqrt 3-1)/(2 sqrt 2)`

(ii) `cos 15^o= ( sqrt 3 +1)/( 2sqrt 2)`

(iii) `tan 15^o = cot 75^o = 2 - sqrt 3`

(iv) `cot 15^o = tan 75^o = 2 + sqrt 3`

(v) `sin 18^o = cos 72^o =(sqrt 5 -1)/4`

(vi) `cos 36^o = sin 54^o =(sqrt 5+1)/4`

(vii) `cos 18^o = sin 72^o =( sqrt(10 + 2 sqrt 5))/4`

(viiii) `sin 36^o = cos 54^o = (sqrt (10 - 2 sqrt 5))/4`

(ix) `sin 22 \ 1/2 ^o= ( sqrt (2 + sqrt 2))/2`

(x) `cos 22 \ 1/2 ^o = (sqrt (2 + sqrt 2))/2`

(xi) `tan 22 \ 1/2 ^o = sqrt 2 -1`

(xii) `cot 22 \ 1/2 ^o = sqrt 2 +1`

The maximum and minimum values of `a sin theta ± bcos theta` are respectively, `sqrt(a^2 +b^2)` and `- sqrt(a^2 +b^2)`

An equation involving one or more trigonometric ratios of unknown angles is called a trigonometric equation.

e.g., `2 cos theta + 3 cos 2 theta =0, cos^2 theta + sin theta= 1/3` etc.

A value of the unknown angle which satisfies the given equation, is called a solution or root of the equation. The trigonometric equation may have infinite number of solutions and can be classified as

(i) Principal solution : The least value of unknown angle which satisfies the given equation, is called principal
solution of trigonometric equation.

(ii) General solution : We know that trigonometric functions are periodic and solutions of trigonometric equations can be generalised with the help of the periodicity of the trigonometric functions. The solution consisting of all possible solutions of a
trigonometric equation is called its general solution.