Math ACT Formulas | Perfectmathsat

Advanced ACT Math Formulas

Advanced ACT math formulas, Advanced ACT math formula sheet, Advanced ACT equations

To find $23\%$ of , take $.23 \times 70$ . is what percent of ? $\dfrac{23}{70} \times 100$ .

is $70\%$ of what? $23 = .7x, \dfrac{23}{.7}=x$ .

A $10\%$ increase followed by a $20\%$ increase is $(1.1\times 1.2) - 1\times 100 = 32\%$ .

$\text{Distance} = \text{Rate} \times \text{time}$

Midpoint formula: $\Bigg(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\Bigg)$

Slope formula: $\dfrac{y_1 - y_2}{x_1 - x_2}$

Distance formula: $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 }$

If two lines are perpendicular, their slopes are negative reciprocals.

Given a point and a slope , the equation of the line is

Alternatively, you can use , and solve for

Slope of line in form is $-\dfrac{a}{b}$

In slope intercept form: $by = -ax + c \rightarrow y = -\dfrac{a}{b}x+\dfrac{c}{b}$

Triangle inequality: if two sides have length and , the other side is

A triangle has side lengths with ratios $1-\sqrt{3}-2$

A triangle has side lengths with ratios $1-1-\sqrt{2}$

Volume of a box:

Volume of a cube:

Surface area of a box:

Surface area of a cube:

Diagonal of a box: $\sqrt{l^2+w^2+h^2}$

Diagonal of a cube: $\sqrt{3}s^3$

If a cube is inscribed in a sphere, the diagonal of the cube = the diameter of the sphere.

Volume of a cone: $\dfrac{\pi r^2 h}{3}$

Surface area of a cylinder: $2 \pi r^2 + 2\pi r h$

Surface area of a sphere: $4\pi r^2$

Volume of a sphere: $\dfrac{4\pi r^3}{3}$

Area of an equilateral triangle: $\dfrac{\sqrt{3}s^2}{4}$

Area of a parallelogram:

Area of a trapezoid: $\dfrac{(b_1 + b_2)h}{2}$

Rhombus area formula: $\dfrac{d_1 d_2}{2}$

Perimeter of a rectangle:

Circumference of circle: $2 \pi r$

Area of a circle: $\pi r^2$

Volume of cylinder: $\pi r^2 h$

Surface area of cylinder: $2 \pi r^2 + 2 \pi r h$

The diagonals of a rhombus are perpendicular.

Sum of interior angles of polygon: $180^{\circ}(n-2)$

The sum of the roots of a quadratic: $-\dfrac{b}{a}$ .

The quadratic formula: $\dfrac{-b \pm \sqrt{b^2 - 4ac} }{2a}$

-coordinate of the vertex of a parabola: $-\dfrac{b}{2a}$

is a parabola in vertex form. Its vertex is .

To find the equation of a quadratic given -intercepts and , take and FOIL it out.

Sum and difference of cubes formulas: ,

$\text{cot} = \dfrac{1}{\text{tan}} \quad \text{csc} = \dfrac{1}{\text{sin}} \quad \text{sec} = \dfrac{1}{\text{cos}}$

$\text{sin} = \dfrac{\text{opp}}{\text{hyp}} \quad \text{cos} = \dfrac{\text{adj}}{\text{hyp}} \quad \text{tan} = \dfrac{\text{opp}}{\text{adj}}$

Radians to degrees: $\dfrac{\pi}{180^{\circ}}$

Degrees to radians: $\dfrac{180^{\circ}}{\pi}$

Law of sines: $\dfrac{\text{sin } A}{a} = \dfrac{\text{sin } B}{b} = \dfrac{\text{sin } C}{c}$

Law of cosines: $c^2 = a^2 + b^2 - 2ab \text{ cos } C$ or $\text{cos } C = \dfrac{a^2 + b^2 + c^2}{2ab}$

$\text{sin }^2 x + \text{cos}^2 x = 1, \quad \text{sin}^2 x = 1 - \text{cos}^2 x, \quad \text{cos}^2 x = 1 - \text{sin}^2 x$

$\text{tan }^2 x + 1 = \text{sec}^2 x, \quad 1 + \text{csc}^2 x = \text{cot}^2 x, \quad \text{sec}^2 x - \text{tan}^2 x = 1$

$\text{sin } 2x = 2 \text{sin } x \text{cos } x, \quad \text{cos } 2x = \text{cos}^2 x - \text{sin}^2 x = 2 \text{cos}^2 x - 1 = 1 - 2\text{sin}^2 x$

$\text{sin } 30^{\circ} = \text{cos } 60^{\circ} = \dfrac{1}{2}, \text{sin } 60^{\circ} = \text{cos } 30^{\circ} = \dfrac{\sqrt{3}}{2},$ and

$\text{sin } 45^{\circ} = \text{cos } 45^{\circ} = \dfrac{\sqrt{2}}{2}, \text{tan } 30^{\circ} = \dfrac{\sqrt{3}}{3},$ and $\text{tan } 60^{\circ} = \sqrt{3}$

The trigonometric area formula is $\text{Area} = \dfrac{ab \text{ sin } A}{2}$

In the function $y = a \text{ sin } bx, |a|$ is the amplitude and $\dfrac{2\pi}{b}$ is the period.

$\text{log } ab = \text{log } a + \text{log } b$

$\text{log } \dfrac{a}{b} = \text{log } a - \text{log } b$

$\text{log } a^b = b \text{log } a$

$\text{arc length} = 2 \pi r \dfrac{\theta}{360}$

$\text{area of pie slice} = \pi r^2 \dfrac{\theta}{360}$

Circle with center and radius is

Factorial: $n! = n(n-1)(n-2) \cdots 1$

Combinations: pick of : $\dfrac{n!}{k!(n-k)!}$

Permutations: $\dfrac{n!}{(n-k)!}$

The number of permutations of the letters in MISSISSIPPI is $\dfrac{11!}{(4! \times 4! \times 2!)}$

A sum of arithmetic series formula is $a_1 + a_2 + \cdots + a_n$ is $\dfrac{(a_1 + a_n) \times n}{2}$

The sum of an infinite geometric series: $a_1 + a_2 + \cdots = \dfrac{a_1}{(1-r)}$

for depreciation,

for simple interest, use

for compound interest, use $A = P \Bigg( 1 + \dfrac{r}{n} \Bigg)^{nt}$

for continuous interest, use $A= Pe^{rt}$

Contrapositive: “All tables are flat” is the same as “If it is not a table, it is not flat”. Switch them and make them both not, and it is the same thing.

Rotating $90^{\circ}$ clockwise, $(a,b) \rightarrow (-b,a)$ ,