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THIGONOMETRIC RATIO AND IDENTITIES
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1 Some important result :
(i) Arc length AB = r q Area of circular sector = (1/2) r2 q
(ii) For a regular polygon of side a and number of sides n
(a) Internal angle of polygon = (n-2) (p/n)
(b) Sum of all internal angles = (n-2) p
(c) Radius of incircle of this polygon r = (a/2)cos (p/n)
(d) Radius of circumcircle of this polygon R = (a/2) cosec (p/n)
(e) Area of the polygon = 1/4 na2cot (p/n)
(f) Area of triangle = (1/4) a2 cos (p/4)
(g) Area of incircle = p{(a/2)cot(p/n)}2
(h) Area of circumcircle = p {(a/2)cosec(p/n)}2
2 Relation between system of measurement of angles:
 and p radian = 180°
3 Trigonometric identities :
(i) sin2 q + cos2 q = 1
(ii) cosec2 q - cot2 q = 1
(iii) sec2 q -tan2 q =1
4 Sign Convention:
5 T-ratios of allied angles:
the signs of trigonometrical ration in different quadrant.
| allied Ð of T-rations | (- q) | 90° ± q | 180° ± q | 270° ± q | 360° ± q |
| sin q | -sin q | cos q | ±sin q | -cos q | ±sin q |
| cos q | cos q | ±sin q | -cos q | ±sin q | cos q |
| tan q | -tan q | ±cot q | ±tan q | ±cot q | ±tan q |
| cot q | -cot q | ±tan q | ±cot q | ±tan q | ±cot q |
| sec q | sec q | ±cosec q | -sec q | ±cosec q | sec q |
| cosec q | -cosec q | sec q | ±cosec q | -sec q | ±cosec q |
6 Sum & differences of angles of t-ratios:
(i) sin (A±B) = sinA cosB ± cosA sinB
(ii) cos(A±B) = cosA cosB ± sinA sinB
(iii) tan (A±B) 
(iv) cot (A±B) 
(v) sin(A+B) sin(A-B) = sin2A – sin2B = cos2B – cos2A
(vi) cos(A+B) cos(A-B) = cos2A – sin2 B = cos2B – sin2A
(vii) tan(A+B+C)
Generalized tan (A+B+C+ ……. )
Where S1 = åtan A
S2 = åtan A tan B,
S3 = åtan A tan B tan C and so on
(viii) sin (A+B+C) = å sin A cos B cos C – Õ sin A = Õ cos A (Numerator of tan (A+B+C))
(ix) cos (A+B+C) = å sin A cos B cos C - Õ sin A = Õ cos A (Denominator of tan (A+B+C))
For a triangle A+B+C =p
å tan A = Õ tan A
å sin A = å sin A cos B cos C
1+ Õ cos A = å sin A sin B cos C
(viii) sin75° =  = cos15°
(ix) cos75°  = sin15°
(x) tan75° = 2 + Ö3 = cot15°
(xi) cot75° = 2 - Ö3 = tan15°
7 Formulaes for product into sum or difference and viceversa:
(i) 2sinA cosB = sin(A+B) + sin (A-B)
(ii) 2cosA sinB = 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) sinC + sinD = 2sin 
(vi)sinC – sinD = 2cos 
(vii) cosC – cosD = 2cos 
(viii) cosC – cosD = 2sin 
(ix) tanA + tanB = 
8 T-ratios of multiple and submultiple angles:
(i) sin2A = 2sinA cosA = 
= (sinA + cos A)2 - 1 = 1 (sin A – cos A)2
Þ sinA = 2sinA/2 cosA/2 = 
(ii) cos2A = cos2A - sin2A = 2cos2A – 1 = 1- 2sin2A = 
(iv) sin3q = 3sinq - 4sin3q = 4sin(60° -q ) sin(60° + q) sinq = sinq (2 cos q -1 ) (2 cos q + 1)
(v) cos3q = 4cos3q - 3cosq
=4cos(60° - q) cos (60° + q)cosq
=cos q (1-2 sin q) (1+2 sin q)
(vi) tan3A =
= yan (60° - A ) tan(60° + A) tanA
9 Maximum and minimum value of the expression:
acosq + bsinq
maximum (greatest) value = Ö(a2+b2)
minimum (least) value = - Ö(a2+b2)
10 Conditional trigonometric identities :
If A,B,C are angles of triangle i.e. A+B+C = p, then
(i) sin2A + sin2B + sin2C = 4sinA sinB sinC i.e. å sin 2A = 4 Õ (sinA)
(ii) cos2A + cos2B + cos2C = -1 – 4cos A cosB cosC
(iii) sinA + sinB + sinC = 4cos A/2 cosB/2 cosC/2
(iv) cosA + cosB + cosC = 1+ 4 sinA/2 sinB/2 sinC/2
(v) sin2A + sin2B + sin2C = 1-2sinA sin B sinC
(vi) cos2A + cos2B + cos2C = 1 -2cosA cosB cosC
(vii) tanA + tanB + tanC = tanA tanB tanC
(viii) cotB cotC + cotC cotA + cotA cotB = 1
(ix) å tan A/2 tan B/2 = 1
(x) å cot A cot B =1
(xi) å cot A/2 = Õ cot A/2
11 Some useful series:
(i) sina + sin(a + b) + sin(a + 2b) + …….. + to nterms
(ii) cosa + cos (a + 2b) + cos (a + 2b) + …… + to nterms
(iii) cosa. Cos2a. Cos22a …….. cos(2n-1 a) =  ,
a ¹ np = 1, a = 2kp = - 1, a = (2k+1)p
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