all topics covered

TRIGONOMETRIC FUNCTIONS
CONCEPTS AND RESULTS
Angles : Angle is a measure of rotation of a given ray about its initial point.
** Measurement of an angle.
**English System (Sexagesimal system)
(i) 1 right angle = 90 degrees = 90o. (ii) 1o = 60 minutes = 60‟. (iii) 1‟ = 60 second = 60‟‟.
**French System (Centesimal system)
(iv) 1 right angle = 100 grades = 100 g. (v) 1 g = 100 minutes = 100 „ (vi) 1‟ = 100 seconds =
100 „‟
**Circular System.
(vii) 180o = 200 g =  radians = 2 right angles, where a radian is an angle subtended at the centre of a
circle by an arc whose length is equal to the radius of the circle.
(viii) The circular measure  of an angle subtended at the centre of a circle by an arc of length l is
equal to the ratio of the length l to the radius r of the circle.
2n4
(ix) Each interior angle of a regular polygon of n sides is equal to right angles.
n
    2 3 5
T-ratios 0 
6 4 3 2 3 4 6
1 1 3 3 1 1
Sin 0 1 0
2 2 2 2 2 2
3 1 1 1 1 3
Cos 1 0 – – – –1
2 2 2 2 2 2
1 1
tan 0 1 3 n.d – 3 –1 – 0
3 3


** Formulae for t-ratios of Allied Angles :
 3
All T-ratio changes in   and   while remains unchanged in    and 2    .
2 2
   3  
sin     cos sin     = cos 
2   2  2
   3 
cos      sin  cos      sin  II Quadrant I Quadrant
2   2 
   3 
tan       cot  tan       cot  sin  > 0 All > 0
2   2 
sin     sin  sin2     sin   0
cos    = = cos  cos2    cos tan  > 0 cos  > 0
tan    =  tan  tan2     tan 
III Quadrant IV Quadrant
3
** Sum and Difference formulae :
2
sin(A + B) = sin A cos B + cos A sin B
sin(A – B) = sin A cos B – cos A sin B
cos(A + B) = cos A cos B – sin A sin B
cos(A – B) = cos A cos B + sin A sin B


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, tanA  tanB tanA  tanB   1  tan A
tan(A + B) = , tan(A – B) = . tan   A   ,
1 - tanA tanB 1  tanA tanB 4  1  tan A
  1  tan A cot A. cot B  1 cot A. cot B  1
tan   A   cot(A + B) = cot(A – B) =
4  1  tan A cot B  cot A cot B  cot A
sin(A + B) sin(A – B) = sin A – sin B = cos2B – cos2A
2 2

cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A

**Formulae for the transformation of a product of two circular functions into algebraic sum of
two circular functions and vice-versa.
2 sinA cos B = sin (A + B) + sin(A – B)
2 cosA sin B = sin (A + B) – sin(A – B)
2 cosA cos B = cos (A + B) + cos(A – B)
2 sinA sin B = cos (A - B) – cos(A + B)

CD CD CD CD
sin C + sin D = 2 sin cos , sin C – sin D = 2 cos sin .
2 2 2 2
CD CD CD CD
cos C + cos D = 2 cos cos , cos C – cos D = – 2 sin sin .
2 2 2 2
** Formulae for t-ratios of multiple and sub-multiple angles :
2 tan A
sin 2A = 2 sin A cos A = .
1  tan 2 A
2 2 2 1  tan 2 A
2
cos 2A = cos A – sin A = 1 – 2 sin A = 2 cos A – 1 =
1  tan 2 A
A A
1 + cos2A = 2cos2A 1 – cos2A = 2sin2A 1 + cosA = 2 cos2 1 – cosA = 2 sin 2
2 2
2 tan A 3 tan A  tan 3 A
tan 2A = , tan 3A = .
1  tan 2 A 1  3 tan 2 A
sin 3A = 3 sin A – 4 sin3A, cos 3 A = 4 cos3A – 3 cos A
3 1 3 1
sin15o = cos75o = . & cos15o = sin75o = ,
2 2 2 2
3 1 3 1
tan 15o = =2– 3 = cot 75o & tan 75o = =2+ 3 = cot 15o.
3 1 3 1
5 1 5 1
sin18o = = cos 72o and cos 36o = = sin 54o.
4 4
10  2 5 10  2 5
sin36o = = cos 54o and cos 18o = = sin 72o.
4 4
o o o
 1 1
o
 1  1
tan  22  = 2 – 1 = cot 67 and tan  67  = 2 + 1 = cot  22  .
 2 2  2  2
** Properties of Triangles : In any  ABC,
a b c
  [Sine Formula]
sin A sin B sin C
b2  c2  a 2 c2  a 2  b2 a 2  b2  c2
cos A = , cos B = , cos C = .
2 bc 2 ca 2 ab
** Projection Formulae : a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A



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