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SOLID GEOMETRY:,
PROPERTIES O FPLATONIC SOLIDS;
Tetrahedron = a squared-square root of 3,
Hexahedron = 6-a squared,
Octahedron = 2-a squared-squareroot of 3,
Prism and Cylinder;
Volume = Ab-h,
Lateral area = Pr-l, pr is perimeter of right section,
Right circular cylinder;
Volume = Ab-h or pie r squared – h,
Lateral area = 2 pier-h,
Pyramid and cone;
Volume = one third Ab-h,
Right circular cone;
Volume = one third pie r squared- h,
Lateral area = pie r-l,
Slant height = squareroot of x squared plus h squared,
Pyramids;
Properties of pyramid;
Ab over Ay = h squared over y squared,
Regular pyramids;
Area of one lateral face = one half x-l,
Lateral area = PL over 2,
Length of lateral edge, e = squareroot of r squared plus h squared,
Slant height,L = squareroot of r squared plus h squared,
Volume = one third Ab –h, or pie over six , r squared – h sin theta,
Truncated prism;
Volume = area of the right section times h-1 plus h-2 plus h-3 over number oh h,
Frustum;
Volume = one third (A-1 plus A-2 plus squareroot of A-1. A2 )- h,
Lateral area = one half (p-1 plus p2 )-L,
Sphere;
Surface area = 4 pie r squared, or pie D squared,
Volume = four third pie r cubed,
Spherical zone;
Area of the zone = circumference of the great circle times altitude,
Area = 2 pie r –h,
Spherical sector;
Total surface area = 2 pie r- h plus pie a-R plus pie b-R or,
Pie r ( 2h+a+b)
Volume = two third pie r squared-h,
Spherical segment:
Total area = 2 pie r –h plus pie a squared plus pie b squared,.
Volume of spherical segment of two bases given = one six. Pie –h (3a squared plus 3b squared plus h squared,.
Volume of spherical segment of one base given = one third pie – h squared ( 3r minus h)
Spherical wedge and spherical lune;
Volume of wedge over theta = four third pie r cubed over 360 degrees,
Area of lune over theta = 4 pie r squared over 360 degrees,.
Similar figures;
Any kind of pairs of corresponding line segments ,x-1,x-2 and y-1 , y-2 have the same ratio = x-1 over x-2 = y-1 over y-2,.
Areas of similar surfaces,A-1 and A-2 have the sam ratio =A-1 over A-2 = (x-1 squared) over (x-2 squared ),
Volume of similar solid V-1 and V-2 have tha same ratio = V-1 over V-2 = (x-1 cubed ) over (x-2 cubed),
Area of triangle :
One base and the altitude = one half b-h,
Two base and included angle = one half a-b sin C, or one half a-c sin B,
Given three sides (heros formula ) = squareroot of s (s minus a),(s minus b), (s minus c )
S=perimeter , = a+b+s over 2.,
c
centers of triangle;
radius of incircle, r = Area of triangle over S,
radius of circumcenter, r = abc over 4 Area of triangle,
quadrilateral :,
perimeter = a+b+c+d,
area = squareroot of , (s minus a), (s minus b), (s minus c), (s minus d) – abcd cos squared theta,.
Area can also be expressed in terms of diagonal d-1 and d-2,
Area = one half d-1. D-2 sine theta,.
Common quadrilaterals;
Rhombus a parallelogram with four equal sides,.
Given diagonals d-1 ad d-2 :,
Area = one half d1-d2,.
Given side a and one angle A = a squared sine theta,.
Area of Parallelogram ;
Given diagonals d1 and d2 and included angle theta = one half d1 –d2 sine theta,.
Given two sides a and b and one angle = ab sine A,.
Trapezoid;
Area = a+b over 2 – h,.
Cyclic quadrilaterals a quadrilateral whose vertices lie on the circumference of a circle:
Area = squareroot of (s minus a), (s minus b), (s minus C ), (s minus d),.
D1-d2 = ac + bd ,
Regular polygons;
Area of one segment = one half r squared sine theta,
Total area = n over 2 ( r squared sine theta,.)
Perimeter = n x.,
Angle = 360 over n,
Circle;
Area of circle = pie r squared,
Circumference = 2 pie r, or pie d,.
Length of arc , S = pie r theta over 180 degrees
Area of sector = pie r squared theta over 360 degrees,.
Radius of circle:
Radius of circle circumscribed about a triangle = abc over 4 Area of triangle,
Radius of circle inscribed in a triangle = area of triangle over s,
Circle escribed about a triangle, (excircle)
Radius = area of triangle over ,(s minus a)
Circle cimcumscribed about a uadrilateral;
Radius = squareroot of (ab +cd ), (ac + bd ), (ad +bc ), over 4 area of quadrilateral,.
Area of quadrilateral = squareroot of (s minus a), (s minus b), (s minus c), (s minus d),
Circle inscribed in a quadrilateral , radius =area of quadrilateral over s,.
Area of quadrilateral = squareroot of abc,.
Quadratic formula :
Quadratic equation : Ax squared + Bx + C = 0,
X= -b plus minus squareroot of b squared minus 4 ac over 2a,
Where; B squared minus 4 ac called discriminant
If b squared minus 4 ac = 0,. The roots are equal,
If b squared minus 4 ac greater than 0, the roots are real , un equal,.
If b squared minus 4 ac less than 0,the roots are imaginary,
Properties of roots :
Sum of roots, x1 + x2 = negative B over C,
Product of roots, x1-x2 = c over a,
Binomial theorem : in the expansion of (x+y ) raised to n,
Rth term = n ! over ( n-r+1) ! (r-1 )! Times x raised to (n-r+1) times y (r-1)
Middle term = n over 2 plus 1,.
Arithmetic progression :
A-n = a1 + (n minus 1)-d ,
Sum of term = n over 2 (2 a1 + (n minus 1 )-d,.
D= d2 minus d1,
Geometric progression:
A-n = a1-r raised to (n minus 1)
R=a2 over a1
Sum of term , when r is greater than 1, = a1 (r raised to n minus 1) over r minus 1)
When r is less than, a1 (1-r raised to n) over r minus 1)
Sum of infinite geometric progression = a1 over 1 minus r,
Sign of function values;
ALL STUDENT TAKE CALCULUS (POSITIVE VALUE)
SUM OF DIFFERENCE OF TWO ANGLE:,
Sine (x+y) = sine x-cosine y + cosine x sine y),
Sine (x-y ) = sine x – cos y minus cosine x – sin y)
Cosine (x+y ) = cosine x cosine y minus sin x-sin y,
Cosine (x-y ) = cosine x cosine y plus sin x-sin y,
Tan (x +y ) = tan x + tan y over 1 plus tanx-tany,
Tan (x -y ) = tan x + tan y over 1 minus tanx-tany,
Double angle identity :
Sine 2x = 2 sine x-cos x,
Cosine 2x = cos squared x minus sin squared x,.
Tangent squared x = 2 tangent x over 1 minus tangent squared x,.
Power of function:
Sine squared x = 1 minus cosine 2x, over 2,.
Cosine squared x = 1 plus cosine 2x, over 2,.
Tangent squared x = 1 minus cosine 2x, over 1 plus cosine 2x,.
Sine law:
A over sine A = b over sine B
Cosine law:
A squared = b squared + c squared – 2bc cosine A,.
Spherical triangle:
A+B+C =180 DEGREES,.
Area of spherical triangle = pie r squared – E, over 180 degrees,.
E = A+B+ C – 180 Degrees,.
Oblique spherical triangle:
Sine law :
Sine a (small) over sine A(big) = sine b (small) over sine B (big)
Cosine law for sides :
Cos –cos .cos + sin.sin.cos,
Cosine a = cosine b-cosine c + sine b-sine c, cosine A (big)
Cosine law for angles : ,
Cosine A = negative cosine B-cosine C + sine C-sine C, cosine a (small)
amplitude= the absolute value,
period = 2 pie over the absolute value. ( in parenthesis),
frequency is closely related to period, it is the reciprocal of the period,.
sample:for amplitude:
y=10sine x, answer; 10,
period and frequency sample;
y= sine 3x, answer for frequency is 3, period = 2 pie over 3.,
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