TRIGONOMETRY FORMULAE
Measurement of Angles
Sexagesimal system of angles; 1 rt. Angle = 90 degree = 90º, 1º = 60 minute = 60/ , 1/ = 60
seconds = 60//
Contesimal system of angles ; 1 rt. Angle = 100 g ; 1g = 100/ ; 1/ = 100//
Circular system; 1 radian = ( 2rt. Angles) / π
Relation
1 radian = 57º 17 /44.8 // nearly 1º = 0. 0 1745 radians only 1 r = 63 g 66 / 20 / / nearly ;
1 g =( 9/ 10)º & 1º = ( 10/9)g ;
180º = 200 g = π radians = 2 rt. Angles
θ = l / r ; θ = angle at centre subtended by are of length ‘l’ and radius of circle = r
BASIC FORMULAE
1. Sin2 + cos 2 = 1 ; Sin2 =1 - cos2 ; 1 - Sin² = cos²
2. 1+ tan2 = sec2 ; sec² – tan2 = 1,
3. 1+ cot2 = cosec2 ; cosec² – cot2 =1
4. sin . cosec =1 ; cos . sec = 1 ; tan . cot =1
5. –1 ; –1 ; sec OR sec ; cosec OR cosec ;
TRIGNOMETRIC FUNCTIONS OF STANDARD ANGLES
00 300 = ( 450 600 900
) ( ) ( ) ( )
Sine 0 1
Cosine 0
1
Tangent 0 1 Not defined
Cosecant Not 2 1
defined
Secant 2 Not defined
1
Cot Not 1 0
defined
ALLIED – ANGLES & SIGNS OF T-RATIOS
Sine & Cosecant (+ve) All T-Ratios ( +ve )
(180 - ) , ( 90 + ) (360 + ),
Tangent & Cotangent( + ve) Cosine & Secant ( + ve )
( 180 + ) , ( 270 - ) (270 - ) , ( 360 - ),(- )
Sine Cosine Tangent Cosecant Secant Cot
90 - cos sin cot sec Cosec tan
90 + cos - sin - cot sec - cosec - tan
180 - sin - cos - tan cosec - sec - cot
180+ - sin - cos tan - cosec - sec cot
270 - - cos - sin cot - sec - cosec tan
270+ - cos sin - cot - sec Cosec - tan
360 - - sin cos - tan - cosec Sec - cot
360+ sin cos tan cosec Sec cot
- - sin cos - tan - cosec Sec - cot
For T-Ratios , if allied angle is , ; then Sin , Tan , Sec
If allied angle is , & - , then NO CHANGE OF T-RATIO.
Addition and subtraction formulae
1) sin (A + B) = sin A cos B + cos A sin B
4) cot ( A + B) =
2) cos ( A+ B) = cos A cos B – sin A sin B
3) tan ( A+ B) =
5) sin ( A- B) = sin A cos B – cos A sin B
6) cos ( A- B) = cos A cos B + sin A sin B
7) tan ( A- B) = 8) cot (A – B) =
9) sin ( A + B) . sin ( A – B) = sin ²A - sin ²B = cos² B – cos² A
10)cos (A + B) . cos ( A- B) = cos² A – sin² B = cos² B – sin² A
T-RATIOS OF SPECIAL ANGLES
150 180 360
7 22
Sine
Cosine
Tangent 2-
Sum & Differences into Products
1) sin C + sin D = 2 sin cos 4) cos C – cos D = -2 sin sin
2) sin C – sin D = 2 cos sin
3) cos C + cos D = 2 cos cos
5) tan A + tan B = ; cot A + cot B =
6) tan A – tan B = ; cot A – cot B = -
7) 2 sin A cos B = sin ( A+B) + sin ( A-B)
2 cos A sin B = sin ( A+ B) – sin ( A – B)
2 cos A cos B = cos ( A+ B) + cos ( A-B)
2 sin A sin B =cos( A-B) – cos ( A+B)
Multiple angles
1) sin 2A = 2 sin A cos A = ; sin A = 2 sin cos =
2) cos 2A = cos² A – sin² A = 2 cos² A – 1 = 1- 2 sin² A=
3) tan 2A = ; tan A =
4) sin 3A = 3 sin A – 4 sin³ A ; cos 3A = 4 cos³ A – 3 cos A
5) Tan 3A =
6) 1- cos A = 2 ; 1 + cos A = 2
7) = tan² ; = cot ²
