20.
TRIGNOMETRIC RATIOS & IDENTITIES
KEY CONCEPTS
1. MEASUREMENT OF ANGLES : There are three systems of measurement of angles.
(i) Sexagesimal or English System(degrees) : Here 1 right angle = 90° (degrees)
1° = 60' (minutes)
1' = 60" (seconds)
(ii) Centesimal or French System(grades) : Here 1 right angle = 100 g (grades)
1g = 100' (minutes)
1' = 100" (seconds)
(iii) Circular system : Here an angle is measured in radians. radian = 180°
If is the angle subtended at the centre of a circle of radius 'r',
by an arc of length 'l' then . • r
r
D G R
Relation between the three systems : 90 100 / 2
2. T-RATIOS (or Trigonometric functions) :
p b p h
In a right angle triangle sin ; cos ; tan ; p
h h b
b b
h h
cosec ; sec q = and cot q p
'p' is perpendicular ; 'b' is base and 'h' is hypotenuse.
p b
3. DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS :
T-Ratio Domain Range Period
sin x R [–1,1] 2
cos x R [–1,1] 2
tan x R–{(2n+1) /2 ; n I} R
cot x R–{n : n I} R
sec x R– {(2n+1) /2 : n I} (– ,–1] U [1, ) 2
cosec x R– {n : n I} (– ,–1] U [1, ) 2
4. BASIC TRIGONOMETRIC IDENTITIES :
(i) sin . cosec = 1 (ii) cos . sec =1
sin cos
(iii) tan . cot = 1 (iv) tan & cot
cos sin
(v) sin 2 + cos2 = 1 (vi) sec2 – tan 2 = 1
(vi) cosec2 cot2 = 1
Concepts of Maths..... 73
5. SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS :
90°, / 2
II quadrant I quadrant
only sine All +ve
& cosec +ve
180°, 0°, 360°, 2
only tan & cot only cos
+ve & sec +ve
III quadrant IV quadrant
270°, 3 / 2
6. TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES :
If is any angle, then , 90 ± , 180 ± , 270 ± , 360 ± etc. are called Allied Angles .
(i) sin ( ) = sin cos ( ) = cos
(ii) sin (90°- ) = cos cos (90° ) = sin
(iii) sin (90°+ ) = cos cos (90°+ ) = sin
(iv) sin (180° ) = sin cos (180° ) = cos
(v) sin (180°+ ) = sin cos (180°+ ) = cos
(vi) sin (270° ) = cos cos (270° ) = sin
(vii) sin (270°+ ) = cos cos (270°+ ) = sin
7. VALUES OF T-RATIOS OF SOME STANDARD ANGLES :
Angles 0° 30° 45° 60° 90° 180° 270°
T-ratio 0 /6 /4 /3 /2 3 /2
sin 0 1/2 1/ 2 3/2 1 0 –1
cos 1 3/2 1/ 2 1/2 0 –1 0
tan 0 1/ 3 1 3 N.D. 0 N.D.
cot N.D. 3 1 1/ 3 0 N.D. 0
sec 1 2/ 3 2 2 N.D. –1 N.D.
cosec N.D. 2 2 2/ 3 1 N.D. –1
Note :
N.D. Not Defined
(i) sin n = 0 ; cos n = (–1) n ; tan n = 0 where n I
(ii) sin(2n+1) = (–1) n ; cos(2n+1) = 0 where n I
2 2
Concepts of Maths..... 74
8. TRIGONOMETRIC RATIOS OF THE SUM & DIFFERENCE OF TWO ANGLES :
(i) sin (A + B) = sin A cos B + cos A sin B. (ii) sin (A – B) = sin A cos B – cos A sin B.
(iii) cos (A + B) = cos A cos B – sin A sin B (iv) cos (A – B) = cos A cos B + sin A sin B
tan A tan B tan A tan B
(v) tan (A + B) = (vi) tan (A – B) =
1 tan A tan B 1 tan A tan B
Some more results :
(a) sin (A + B). sin(A – B) = sin 2 A – sin 2 B = cos2 B – cos2 A.
(b) cos (A+B).cos (A – B) = cos 2 A – sin 2 B = cos²B sin²A
9. FACTORISATION OF THE SUM OR DIFFERENCE INTO PRODUCT
C D C D C D C D
(i) sinC + sinD = 2 sin cos (ii) sinC sinD = 2 cos sin
2 2 2 2
C D C D C D D C
(iii) cosC + cosD = 2 cos cos (iv) cosC cosD = 2 sin sin
2 2 2 2
10. TRANSFORMATION OF PRODUCTS INTO SUM OR DIFFERENCE
(i) 2 sinA cosB = sin(A+B) + sin(A B) (ii) 2 cosA sinB = sin(A+B) sin(A B)
(iii)2 cosA cosB = cos(A+B) + cos(A B) (iv) 2 sinA sinB = cos(A B) cos(A+B)
11. TRIGONOMETRIC RATIOS OF SUM OF MORE THAN TWO ANGLES :
(i) sin (A+B+C) = sinAcosBcosC + sinBcosAcosC + sinCcosAcosB – sinAsinBsinC
= sinA cosB cosC – sin A (Not important)
(ii) cos (A+B+C) = cosA cosB cosC – sinA sinB cosC – sinA cosB sinC – cosA sinB sinC
= cos A – sin A sin B cos C (Not important)
tan A tan B tan C tan A tan B tan C S1 S3
(iii) tan (A + B+ C)
1 tan A tan B tan B tan C tan C tan A 1 S2
12. TRIGONOMETRIC RATIOS OF MULTIPLE ANGLES :
(a) Trigonometrical ratios of an angle 2 in terms of the angle :
2 tan
(i) sin 2 = 2 sin cos =
1 tan2
1 tan2
(ii) cos 2 = cos2 – sin 2 = 2 cos2 – 1 = 1 – 2 sin 2
1 tan2
(iii) 1 + cos 2 = 2 cos2 (iv) 1 – cos2 = 2 sin 2
1 cos 2 sin 2 2 tan
(v) tan (vi) tan 2
sin 2 1 cos 2 1 tan2
(b) Trigonometrical ratios of an angle 3 in terms of the angle :
(i) sin3 = 3sin – 4sin 3 = 4 sin sin (60° – ) sin (60° + )
(ii) cos3 = 4cos 3
– 3cos cos cos (60° – ) cos (60° + ).
3
3 tan tan
(iii) tan 3 = tan tan (60° – ) tan (60° + )
1 3 tan 2
Concepts of Maths..... 75
13. TRIGONOMETRIC RATIOS OF SOME STANDARD ANGLES :
3 1 3 1
(i) sin 15 sin co s 75 (ii) cos 1 5 co s sin 7 5
12 2 2 12 2 2
5 1 5 1
(iii) s in 1 8 s in co s 7 2 (iv) cos 36 cos sin 54
10 4 5 4
5
(v) tan 15 tan 2 3 co t 75 (vi) tan 75 tan 2 3 cot15
12 12
3
(vii) tan 22.5 tan 2 1 cot 67.5 (viii) tan 67.5 tan 2 1 cot 22.5
8 8
14. CONDITIONAL TRIGONOMETRIC IDENTITIES :
If A + B + C = 180°, then
(i) tan A + tan B + tan C = tan A tan B tan C (ii) cot A cot B + cot B cot C + cot C cot A = 1
A B B C C A A B C A B C
(iii) tan tan tan tan tan tan 1 (iv) cot cot cot cot cot cot
2 2 2 2 2 2 2 2 2 2 2 2
(v) sin 2A + sin 2B + sin 2C = 4 sinA sinB sinC
A B C
(vi) sin A + sin B + sin C = 4 cos cos cos
2 2 2
(vii) cos 2A + cos 2B + cos 2C = –1 – 4 cosA cosB cosC optimal
A B C
(viii) cos A + cos B + cos C = 1 + 4 sin sin sin
2 2 2
15. MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC FUNCTIONS:
(i) Min. value of a 2tan 2 + b2cot2 = 2ab where R (Using A.M G.M.)
(ii) Maxima and Minima value of acos + bsin are a 2 b 2 and – a 2 b 2
(iii) If A, B, C are the angles of a triangle then maximum value of sinA + sinB + sinC or
cosA + cosB + cosC and sinA sinB sinC occurs when A = B = C = 60 0
(iv) In case a quadratic in sin or cos is given then the maximum or minimum values can be
interpreted by making a perfect square.
16. SUM OF SINES OR COSINES SERIES OF N ANGLES
n
sin 2 n1
sin + sin ( + ) + sin ( + 2 ) + ...... + sin n 1 = sin
sin 2
2
n
sin 2 n1
cos + cos ( + ) + cos ( + 2 ) + ...... + cos n 1 = cos
sin 2 2
Important results :
sin(2 n )
cos cos 2 cos 4 .... cos (2 n–1
)=
2 n sin
Concepts of Maths..... 76
