where: A is the area and x is the side of the square
Area of a Rectangle
A = xy
where: x and y are the adjacent sides of the rectangle
Area of a Circle
A = pi x r2
where: r is the radius of the circle and pi = 3.14
Area of a Triangle
There are three formulas:
First:
A = 1/2 x b x h
where: b is the base of the triangle and h is the height which the
perpendicular distance from the vertex to the base
Second
A = 1/2 ab Sin C
where: a and b are two sides and C is the angle opposite to the
third side c.
Third
A = Ö(s(s-a)(s-b)(s-c)),
Where a, b and c are the three sides of the triangle
and s = (a + b + c)/2 or half the perimeter
Area of Trapezium
[(x+y)/2]h
where: x and y are the two parallel sides and h is the perpendicular
distance between the parallel sides
Area of a Parellelogram
b x h
where: b is the base and h is the perpendicular distance of the
opposite side from the base
VOLUMES IN GEOMETRY
Volume of a Cube
V = x3
where: V is the volume and x is the side of the cube
Volume of a Cuboid
V = xyz
where: x, y and z are the adjacent three sides of the cuboid
Volume of a Sphere
V = (4/3) pi x r3
where: r is the radius of the sphere and pi = 3.14
Volume of a Hemisphere
V = (2/3) pi x r3
where: r is the radius of the hemisphere and pi = 3.14
Volume of a Cylinder
V = pi x r2 x h
where: r is the radius of the cylinder and h is its height and pi = 3.14
Volume of a Rectagular Pyramid
(L x w x h)(1/3)
where: L and w are the two sides of the rectangular pyramid base and h is the
height
Volume of a Cone
(pi x r2 x h)/3
where: r is the radius of the base and h is the height of the prism
SURFACE
AREA IN GEOMETRY
Surface Area of a Cube
SA = 6x^2
where: SA is the surface area and x is the side of the cube
Surface Area of a Cuboid
SA = 2(xy+yz+zx)
where: x, y and z are the adjacent three sides of the cuboid
Surface Area of a Sphere
SA = 4 pi r2
where: r is the radius of the sphere and pi = 3.14
Surface Area of a Cone
SA = pi x r x L
where: r is the radius of the cone and L is the slant height of the cone and
pi =3.14
Total Surface Area of a Cone (including the area of the base)
SA = (pi x r x L) + pi x r2
where: r is the radius of the cone and L is the slant height of the cone and
pi =3.14
PROPERTIES OF TRIANGLES
Pythagorus Theorem
a2 + b2 = c2
Where: c is the hypotenuse of a right angle triangle and a and b are two sides
containg the right angle.
Cosine Law
c2 = a2 + b2 - 2ab (Cos C)
This formula is applicable to any tirangle. Here a, b and c are the
three sides of the triangle and A, B and C are the angle opposite to these sides
respectively
similarly
a2 = b2 + c2 - 2bc (Cos A)
and
b2 = c2 + a2 - 2ca (Cos B)
Note: This is a general law and if you make any of the angles equal
to 90 degrees it gives the Pythagorus Theorom as Cos of 90 degrees = 0
Sine Law
(Sin A)/a = (Sin B)/b = (Sin C)/c
This formula is applicable to any tirangle. Here a, b and c are the
three sides of the triangle and A, B and C are the angle opposite to these sides
respectively
TRIGONOMETRY:-
IDENTITIES IN TRIGONOMETRY
A right angled triangle has a hypotenuse, base and the pependicular side. Let
angle A be the angle between the base and the hypotenuse.
Then
Sin A = perpendicular / hypotenuse
Cos A = base / hypotenuse
Tan A = (Sin A) / (Cos A) = perpendicular / base
Cot A = 1 / Tan A = (Cos A) / (Sin A)
Sec A = 1 / Cos A
Cosec A = 1/ Sin A
Certain basic Trignometry identities applicable to any angle are:
Sin (-A) = - Sin A
Cos (-A) = Cos A
Tan (-A) = - Tan A
Sin2A + Cos2A = 1
1 + Tan2A = Sec2A
1 + Cot2A = Cosec2A
TABLE
IN TRIGONOMETRY
Angles(deg)>>
0
30
45
60
90
Sin A
0
1/2
1/SQRT 2
(SQRT 3)/2
1
Cos A
1
(SQRT 3)/2
1/SQRT 2
1/2
0
Tan A
0
1/(SQRT 3)
1
(SQRT 3)
undefined
Cot A
undefined
(SQRT 3)
1
1/(SQRT 3)
0
Cosec A
undefined
2
SQRT 2
2/(SQRT 3)
1
Sec A
1
2/(SQRT 3)
SQRT 2
2
undefined
Note:
Tan A = (Sin A)/(Cos A)
Cot A = (Cos A)/(Sin A)
Cosec A = 1/(Sin A)
Sec A = 1/(Cosec A)
If we remember these and know the values of Sin and Cos A we can derive the
rest
INDEFINITE INTEGRAL
If 'f' and 'g' are functions of 'x', such that g'(x)=f(x) then the function
'g' is called an integral of 'f' with respect to 'x', and is written
symbolically as:
òf(x)dx = g(x) + c
where: f(x) is called the integrand and 'c' is called the constant of
integration
Note: If d/dx f(x) = g(x) then d/dx {f(x) + c} = g(x)
Where 'c' is constant, because differentiation of a constant is zero.
Thus the general value òg(x)dx is
f(x)+c, where 'c' is the constant of integration.
Clearly integral will change if 'c' changes. Thus integral of a function is
not unique, hence it is called indefinite integral.
Standard Results:
These standard results for integral calculus are derived directly from the
standard results of differential calculus
Differential Calculus
Integral
Calculus
d/dx(xn+1/ n+1) = xn
òxn
dx =(xn+1/n+1) + C [n not =1]
d/dx loge|x| = 1/x
ò1/x dx =
loge|x| + c [n= -1]
d/dx ex = ex
òex
dx = ex + c
d/dx ax = ax logea
òax
dx = ax / logea + c [a>0]
d/dx Cosx = - Sinx
òSinx dx =
- Cosx +c
d/dx Sinx = Cosx
òCosx dx =
Sinx + c
d/dx Tanx = Sec2x
òSec2x
dx = Tanx + c
d/dx Cotx = - Cosec2 x
òCosec2x
dx = - Cotx + C
d/dx Secx = Secx.Tanx
òSecx.Tanx
dx = Secx + c
d/dx Cosecx = - Cosecx.Cotx
òCosec.Cotx
dx = - Cosecx + c
d/dx Sin-1x = 1/v(1-x2)
ò1/(1-x2)
dx = Sin-1 + c
d/dx Tan-1x = 1/(1+x2)
ò1/(1+x2)
dx = Tan-1x + c
d/dx Sec-1x = 1/xv(x2 - 1)
ò1/(x2
- 1) dx = Sec-1 x + C
d/dx Sin-1x/a = 1/v(a2 + x2)
òdx/Ö(a2
- x2) = Sin-1x/a + c
d/dx (1/a) tan-1x/a = 1/(x2+a2)
òdx/(x2+a2)
= 1/a tan-1(x/a) +c
d/dx (1/a Sec-1x/a) =1/xv(x2
- a2)
òdx/xÖ(x2-a2)
= 1/a Sec-1x/a +c
d/dx Coshx = Sinhx
òSinh dx =
Coshx + c
d/dx Sinhx = Coshx
òCoshx dx
= Sinhx + c
d/dx Tanhx = Sech2x
òSec2x
dx = Tanhx + c
d/dx Cothx = - Cosech2x
òCosech2x
dx = - Cothx +c
d/dx Sechx = - Sechx.Tanhx
òSechx.Tanhx
dx = - Sechx + c
d/dx Cosechx = - Cosechx.Cothx
òCosechx.Cothx
dx= -Cosechx+c
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