ACT Math

Challenge areas for ACT math including equations, definitions, etc.

logbase(n)=x (logarithmic)

base^x=n (exponential)

y=A sin (B(x-h)) + k AND y= A cos(B(x-h)) +k

• A is the amplitude (when A < 0, the graph will be reflected vertically over the midline

• B is the frequency in the 2pi interval, and the period can be determined by -2pi/B (when B<0, the graph will be reflected horizontally)

• h<0 shifts the function left, h>0 shifts the function right

• k<0 shifts the function (and thus the mean/midline) down, k>0 shifts the function (and thus the mean/midline) up

Double angle identities

• sin(2A)=2sinAcosA

• cos(2A)=cos^2A-sin^2A

Sum of angles of a polygon

(n - 2) * 180degrees

Distance formula (finds distance between two points on a graph)

d=square root of ((x1-x2)² + (y1-y2)²)

Midpoint formula (finds midpoint between two points on a graph)

(Xm, Ym) = ((x1+x2)/2, (y1+y2)/2)

Parallelogram

a flat, four-sided geometric shape where both pairs of opposite sides are parallel

Area of a trapezoid

A=(b1+b2)/2(height)

Area of a rhombus

A=(d1)(d2)/2

Circumference of a circle

Pi(diameter) AND =2pi(radius)

Area of a circle

Pi(radius)^2

Equation of a circle (center = (h,k) and radius = r)

(x-h)² +(y-k)² = r²

Volume of a cylinder

pi(radius)^2(height)

Volume of a rectangular prism

lwh

Volume of a cube

e^3 (e=edge length)

Surface area of a cube

6e^2 (e=edge length)

Volume of a sphere

4/3)pi(radus)^3

Law of sines

sinA/a = sinB/b + sinC/c

Law of cosines

A²=B²+C²-2BCcos(a) (Can also be B² or C²)

Formula for combinations (Use when you select a smaller group of items from a larger set and the order does not matter, EX: Team or Committee Selection: Picking 4 students out of 20 to join a club.)

n! / (r!(n-r)!)

Negative Exponent

Make the original number into a fraction with 1/n^x (n=original number) and (x=exponent)

Fractional exponent

The denominator tells you the root to take, and the numerator tells you the power to raise the base to.

Square roots

You can multiply and divide square roots but cannot add or subtract them unless base numbers are the same.

Inverses

Find by switching the independent and dependent variables (aka x and y) and then solving for the dependent variable again.

Pythagorean Identities

sin^2 + cos^2 =1, 1+tan^2=sec^2, 1+cot^2=csc^2

Unit circle

2pi (1,0), pi/6 ((square root of 3/2), ½), pi/4 ((square root of 2/2)/(square root of 2/2)), and pi/3 (same as pi/6 but x and y are switched)

30-60-90 triangle

short leg of x, a hypotenuse of 2x, and a long leg of x(square root of 3)

45-45-90 triangle

legs of x and a hypotenuse of x(square root of 2)

Cosecant

csc =1/sin (hyp/opp) reciprocal of sin

Secant

sec=1/cos ((hyp/adj) reciprocal of cosine

Cotangent

cot=1/tan (adj/opp) reciprocal of tangent

For independent events

P(A|B)=P(A), P(B|A)=P(B), and P(A and B)=P(A) * P(B)

For dependent events

P(A|B)=P(A and B)/P(B)

Addition Rule

P(A or B)=P(A)+P(B)-P(A and B)

General Multiplication Rule

P(A and B)=P(A)*P(B|A)=P(B)*P(A|B)