CXC MATHS Formula Sheet

Formula Sheet 4 CXC Area and Perimeter Formula Perimeter = distance around the outside (add all sides).

Trigonometric Formula Opposite – side opposite to angle Adjacent –side beside (adjacent) to angle

hyp θ

Hypotenuse- longest side opposite sin θ = hypotenus adjacent cos θ = hypotenus opposite tan θ = adjacent

c

adj a

hyp

opp b Remember: works only on right angle triangles

Pythagorean Theorem Triples: 3,4,5

5,12,13

8,15,17

C is the hypotenuse, a and b are the other sides. Remember: works only on right angle triangles

Coordinate Geometry Formula Volume and Surface Area

Distance Formula:

Midpoint Formula:

Gradient Formula:

Equation of a line Slope-Intercept Method:

y = mx + c

Point-Gradient Method:

Parallel lines have equal slope. Perpendicular lines have negative reciprocal gradients.

Parallel lines

Angle Information Complementary angles - two angles whose sum is 90. Supplementary angles - two angles whose sum is 180.

Corresponding angles are equal. 2=

General Triangle Information Sum of angles of triangle = 180.

3=

7,

4=

Alternate Interior angles are equal.

Measure of exterior angle of triangle = the 6, sum of the two non-adjacent interior angles. The sum of any two sides of a triangle is greater than the third side.

6,

1=

4=

5,

8 3=

5

Alternate Exterior angles are equal. 1= 8, 2= 7 Same side interior angles are supplementary. m 3+m 5=180, m 4+m 6=180

Solving triangles A Polygons

c

Sum of Interior Angles:

b

B a

Sum of Exterior Angles: Each Interior Angle (regular poly):

Each Exterior Angle (regular poly):

Sine rule a b c sin A sin B sin C = = = = or a b c sin A sin B sin C Used when any two sides and their corresponding angles are involved to find one missing side or angle.

Cosine rule a 2 = b 2 + c 2 − 2bc × cos A b 2 = a 2 + c 2 − 2ac × cos B

Quadratic Formula

C

c 2 = a 2 + b 2 − 2ab × cos C

If

Used when three sides and an angle between them are given to find the other side Heron’s Formula Area of a triangle given only the length of the sides A = s( s − a )( s − b)( s − c )

ax 2 + bx + c = 0

then

x=

− b ± b 2 − 4ac 2a

a+b+c 2 Capital letters represent Angles Common letters represent sides where s =

Tangent

Circle Facts

5. Radius to tangent is 90o at point of contact. 6. The tangents to a circle from an external point T are equal in length.

radius

7. Angle between tangent to circle and chord at the point of contact is eqaual to the to the angle in the alternet segment

diameter

2.

segment 1a. O Sector

4.

Diameter = 2× radius

3.

Area of circle = πr2 Circumference of circle = 2πr or πd

θ 360 θ 2 Area of sector = πr × 360 Length of arc = 2πr ×

6. 5.

Area of Segment = Area of sec tor − Area of triangle

θ  1 2   = πr 2 ×  −  r sin θ  360   2 

T

B

7.

Angles in circles A

E D

T

1. a,b,c,d Angle at the center in twice

1b.

1c.

angle at the circumference. 2. Angle formed on the diameter in 90o 3. Angles in the same segment are equal 1d. 4. opposite angles in a cyclic Quadrilateral are supplimentary( add up to 180o)

Transformational Matrices

Matrices Adding or subtracting matrices a b  e f  a ± e b ± f   ÷±  ÷=  ÷ c d   g h  c ± g d ± h Multiplying Matrices  a b   e f   ae + bg af + bh   ÷×  ÷=  ÷  c d   g h   ce + dg cf + dh  Determinant of 2×2 Matrix a b A= ÷ If c d A = ad − cb A singular matrices has a determinant of 0

REFLECTION Multiply matrices by each point to get reflection in

x- axis

y- axis

− 1   0 

1  0 

0  1 

0   − 1 

0  1 

y=-x 1  0 

 0  − 1 

− 1  0  

TRANSLATION

Movement of x in x direction and y in ydirection add matrix

x   y  to eack point to get its image.   ROTATION Multiply by matrices to get rotation of ϴ-degrees clockwise about origin (0,0)

Adjoint of 2×2 matrix a b A= ÷ c d

 cos θ Rθ   sin θ 

− sin θ   cos θ  

 d −b  A adjo int =  ÷  −c a 

− 1 R180   0 

0   − 1 

Inverse of 2×2 matrix

y=x

0 R90  1 

− 1  0  

 0 R270  − 1 

1  0 

Enlargment Multiply each point by scale factor K to get the image of the point for an enlargment from the origin.

Sets

ξ = universal set ∈= is a member of ∉= is not a member of ∪ = union ∩ = intersect ∅ = null set A' = Elements not in set A

a b A= ÷ c d 1 A −1 = × A adjoint A or A −1 =

 d −b  1 × ÷ ad − bc  − c a 

Reverse oppertions + _

− +

sin θ

sin −1 θ

× ÷

÷ ×

cos θ

cos −1 θ

tan θ

tan −1 θ

f ( x)

f −1 ( x )

x

a

ax

Shape

Volume

Surface Area

Cube l×l×l=l3

6l2

1 l l Cuboid lwh

2lw+2hw+2lh

h l w Prism bh + lb + sl + shl h

1 bhl 2

l b Cylinder r

2 2 2πrπ r+ 2hπrh

h or 2πr ( r + h )

Cone 1

π3r 2πr+ πhrs 2

Sphere 4 πr 3 3 2 4πr

©2006 D. Ferguson