1D

TRIGONOM ETRIC RATIOS

I. () sin A PPside to AC
Hyp AB
9 BC Hyp,
(i) cos e Adjside to
Opp.side
Hyp AB
(iii) Tan = Opnside to 0 ÁC Adj.side
Adjside to
cos 1 sin cos ec® = 1
2. sin 0 (ii) cot9= (iii) co sec =
)) tan = sin 0
cos0 sin 9
1
(iv) sec@
=cosesec® =1 (v) Tan: cot ’ cote tan =1
cos
Quadrant Angles
Silver 90-0,360+0
Q1
(sin,cosec) +ve All(+ve) 90 +-0, 180-0
3 Q2
IV
Q3
180+ 0, 270-0
Tea Cups
(tan,cot) tve (cos,sec) +ve 270+0,360-0
Q4
changes as sin > cos, tán 4’ cot, sec cos ec
4. () For 90°+0,270°±0; trigonometric functions
(ii) For 180°±0,360° ±0 no changes
sin(-0) =-sin0 (ii) cos ec(-0) = -cos ec®
5. )
(ii) cos(-0) =+cos0 (iv) sec(-0) =+sec0
(v) Tan(0) =-tan.0 (vi) cot(-0) =-cot e
cos? 0'=1-sine
() sin? +cos?9 =l’ sin? 0=-cos?0=’ Tano= sec² 0-1
(ii) sec 0-tan 0=l»sec?0=1+ tan²0 1
tan = sec+ tan
’ (sec0 + tan 0)(sece- tan 6) =1’ sece-
(iii) cosec9-cot' 1 cos ec°0 =1+cot 0 cot = cos ec'011
- cot0) =1 cosece cot0 =:
’ (cos ec0 +cot0) (cos ec cOs ec+ cot 0
qoCV) IBO(T) 236(3) 368(1D
7. Angle

Sin 2
-- 2
1 0 -1

0 -1 0
Cos
2
1
Tan
1 W3

=0.01745,1' =57.296° (approximateiy)

Periodicity and Extreme values fur.ction with period k (where k is the least positive real value)
If f(x +k) =f(x) then f(x) is called a periodic
1.
cos ecax is 2r/a
2 (a) i) The period of sin ax, cos ax, sec ax,
cos ecx is 2
i) The period of sin x, cos x, sec x,
n/a
b) The period of tan ax, cot ax is
x is
(ii) The period of tan x, cot
3. (a) The period of a cosx+bsin"x is ifa'=b andn is positive even.
2
(b) The period of a cos" x +bsin" x is n is ab and n is positive even.
(c) The period of a cos" x + bsin" x is 2t ifn is odd.
f(3)
4. The period of f (x)±g(x).g(x) is the L.C.M. of the periods of f(x) and g(X).

S. (a) Maximum value of acosx+bsinztC is ctVab

(b) Minimum value of a cos +bsin a +cis c-a' +b?
COMPOUND ANGLES
1. (a) sin (A +B) =sin Acos B+ cos Asin B; (b) sin (A -B) =sin Acos B- cos AsinB
(c) cos(A +B) = cos A cos B-sin Asin B (d) cos(A-B) =cos Acos B+sin Asin B
TanA +TanB TanA-TanB
(e) Tan (A +B) =1-TanATanB () Tan(A-B) = 1+TanATanB
CotBCotA -1 CotBCotA +1
(s) Cot (A +B) = CotB +Cot A (h) Cot (A-B) =CotB-Cot A
2. (a) Sin (A+B)Sin(A-B) =sin' A-sin?B =cos' B- cos' A
(b) Cos{A +B) cos (A -B) =cos'B-sin' A=cos'A-sin' B
3. (a) Sin(A +B+C) =X(sinA cosB cosC)-sin Asin Bsin C
(b) Cos(A +B+C) =cos Acos Bcos C-)cos Asin BsinC
tan A-II tan A XcotA -IIcotA
() Tan (A+ B+C) = (a) cot (A +B+C) =
1-) tan Atan B 1-Xcot Acot B
a) sin1S° =cos750= 22 b) cos15°=sin 75° =B+1
4. .2/2
(c) tan 15° =cot 75° =2-43 (d) cot 15° =tan 75°=2+\3
V3+1
(e) sin 105° =sin 75° = 2V2 (f)cos105 =- cos 75" =3-)
242
(e) tan 105° =- tan 75° =-(2+3) (h) cot 105° =-cot75° =-2-43)
MULTIPLE AND SUBMULTIPLES ANGLES
2 tan A
1. (a)sin 2A =2sin A cos A= 1+ tan' A
sin² A =l- tan'A
(b)cos 2A =cos' A-sin A= 2cos' A-l=B-2 1+ tan A

(c) tan 2A
2 tan A
(d) cot 2A = cot A-1
-tan' A 2cotA
 2tanA
2. (a)sin A=2sin cos A A
1+ tan
2
1-tan² A
2
(b) cos A=cosA-sin=
2 2
2cos-1=1-2sin 2 1+tan?A
2
Cop?A
" 2 tan A
(c) tan A = (a) cotA =
1-tan² 2 2cot 2

3. (a)1+ cos 2A = 2cos² A (b)1+ cosA = 2cos,2 A

2

(c)1-cos 2A =2sin' A (d)l-cos A= 2sin? A

4. For any A eR
1+ cos2A 1-cos 2A
(b)cosA =| (c) tan A=i+ cos2A
(4)sin A= 2

|1+cos A cos A
(c) tan*i+ cos A
$ (a)sin; .2

(b) cos 3A =4cos' A-3cos A

6. (a) sin 3A 3sin A -4sin' A A
3tanA tan A (d) sin A=3sin A 4sin 3
(c) tan 3A = 1-3 tanA
3 tan
A - tanA.
3 3
(f) tan A= 1-3tan2 A
(e) cosA = 4cos-3cos
3 3
3

3cot-cor A
3 3
(8) cot A= 1-3cot?A
3

7. Angle

V10-2/5 WS+1 V10+2/5

4 4
Sin 4 4

10-2/5 5-1
Vho+2/5 4 4 4
Cos 4
Transformations:
1. (a) sin(A+B) +sin(A-B) = 2sin A.cosB (b) sin (A +B) -sin(A-B) =2 cos A.sin B
fe) cos (A +B) +cos (A-B) =2cosA. cos B (a) cos (A +B) - cos (A -B) =-2sin A. sin B
Ae) cos(A -B) - cos(A +B) = 2sin A.sin B
2. (a) sin C+ sin D=2sin (b) sin C-sin D= 2cos

) osC-+ cosD =2c0

(c) co P)
Trigonometric Equations:
1. (a) The general solution,of sin 0=0, tan 9=0 is = nT neZ
(b) cos =0 then general solution is =(2n +1)neZ
2

2. (a) If sin 6=sin athen general solution is =nr+(-1)' a Vne Z,-sas

2
(b) If cos =cosa then general solution is =2nna tne Z, 0sasn

(c) If tan 0= tan a then general solution is = na+a VneZ,-<a<2

3. If sin'0 sin'a, cos 0= cos a, tan" = tan a theni general solution is = nTta, neZ

4. Common solution of Two trigonometric equations is =2n +a, 0 Sa < 2r

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