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PCM
trigonometry
i
The Basics
cos(A+b)= cosacosb-sinasinb
sin(A+B) = sinacosb+ cosasinb
tan(a+b)= tana+tanb
1-tanatanb
(A+B)
from what we know, we can find the values of 15 and 75 degrees as well.
Table
0
0
15
0
30
0
45
0
60
0
90
0
75
0
180
0
270
0
360
0
0
0
0
0
0
0
0
0
1
1
1
-1
-1
1
1/2
1/2
3
2
3
2
3
1
3
1
2
1
2
WHAT IS SIN OR COS?
Meaning of Sin
(0,1)
(1,0)
(-1,0)
(0,-1)
Consider a point P on a circle of radius 1 units such that OP makes and angle with positive X axis in the anticlockwise direction.
The X Co-ordinate is cos
The Y Co-ordinate is sin
Additional Formulae derived
Derived Additional Formulae
sin(180-x)= sinx sin(180+x)= -sinx
cos(180-x)=-cosx cos(180+x)= -cosx
tan(180-x)=-tanx tan(180+x)= tanx
sin(360-x)= -sinx sin(180+x)= sinx
cos(360-x)=cosx cos(360+x)= cosx
tan(360-x)=-tanx tan(360+x)= tanx
sin(-x)= -sinx
cos(-x)= cosx
tan(-x)= -tanx
Funda
Trignometric Ratio
( )
n /2 + O
nPIE/2 Funda
-
determine the quadrant assuming thita is acute. put +/- sign accordingly.
-
if n is odd, it will be converted to its complimentary ratio
-
if n is even it will remain the same raton
Additional Formulae derived
Important formulae
sin(2x)= 2sinxcosx
cos(2x)= cos^2(x)-sin^2(x)
tan(2x)= 2tanx
1-tan^2(x)
(sinx+cosx)^2= 1+sin2x
sinx= 2tanx
1+tan^2(x)
cos(2x)= cos^4(x)-sin^4(x)
2cot(2x)= cotx-tanx
sinxsin(60+x)sin(60-x)= sin3x/4
2sinacosb= sin(a+b)+sin(a-b)
2cosacosb= cos(a+b)+cos(a-b)
Factorization formulae
cosec2x-cot2x= tanx
Factorization Formulae
defactorization
Defatorization Formulae
sina+sinb= sin(a+b/2)cos(a-b/2)
cosa+cosb=
graphs of trignometric ratios
Graphs
y=sinx
y=cosx
y=tanx
Multiple Fundae
Funda
if
-a+b<asinx+bcosx<a+b
2
2
2
2
tanx=
a/b
asinx+bcosx
2
2
(a,b) or (b,a)
Funda
For sinacosa type or multiplication type questions, you can't multiply the ranges as they have the same angle(a) and the angle that makes sin max will not make cos max.
Funda
Funda
For 2 positive numbers A, B the
minimum value of thier sum is
equal to 2 AB
therefore, A+B> 2 AB
This method is only useful if a=b is possible.
if not, then sin A=1 or cos A=1
2
2
atanx + bcotx
2
2
(2 ab , )
Funda
Chain reaction
chain reaction
1
cosxcos(60-x)cos(60+x)= cos3x/4
2
cosxcos2xcos4xcos8x= sin16x
16sinx
Funda
If A+B= n
if n is odd
sina=sinb
cosa=-cosb
tana=-tanb
if n is even
sina=-sinb
cosa=cosb
tana=-tanb
A+B=nPIE
If A-B= n
if n is odd
sina=-sinb
cosa=-cosb
tana=tanb
if n is even
sina=sinb
cosa=cosb
tana=tanb
sin and cosine of an ap
Arthimetic Proggression
sina+sin(a+d)+sin(a+2d)+.............................
sn=sin[ ]
a+(a+d) sin(nd/2)
2 sin(d/2)
cosa+cos(a+d)+cos(a+2d)+..........................
sn=cos[ ]
a+(a+d) sin(nd/2)
2 sin(d/2)
Funda
(x+y+z)
x+y+z ratios
sin(x+y+z)= s1c2c3+s2c3c1+s3c2c1-s1s2s3
cos(x+y+z)= c1c2c3-(s1s2c3+s2s3c1+s3s1c2)
tan(x+y+z)= t1+t2+t3-t1t2t3
1-t1t2-t2t3-t3t1
if tanx+tany+tanz=tanxtanytanz
then x+y+z=n
if tanxtany+tanytanz+tanztanx=1
then x+y+z=n /2 where n is odd
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