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ACT Math & English High-Yield Rules

Essential geometry formulas, algebra equations, and punctuation conventions for ACT prep.

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SOH CAH TOA Definitions

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SOH CAH TOA Definitions

sin = \frac{opp}{hyp}, cos = \frac{adj}{hyp}, tan = \frac{opp}{adj}

Law of Sines

\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}

Law of Cosines

c^{2} = a^{2} + b^{2} - 2 ab cos (C)

Pythagorean Identity

sin^{2} (θ) + cos^{2} (θ) = 1

Equation of a Circle at

(h, k)

(x - h)^{2} + (y - k)^{2} = r^{2}

$ with center $(h,k)$ and radius $r$.

Logarithm Product and Quotient Rules

lo g (x y) = lo g (x) + lo g (y), lo g (x / y) = lo g (x) - lo g (y)

Logarithm Power Rule

lo g (x^{k}) = k lo g (x)

Imaginary Unit

i

Powers

i = - 1, i^{2} = - 1, i^{3} = - i, i^{4} = 1

Matrix Multiplication Dimension Requirement

A matrix of size

m \times k

can multiply a matrix of size

k \times n

, producing an

m \times n

result.

Arithmetic Mean vs. Median

Mean = sum ÷ count. Median = middle value of an ordered list (or average of two middles if even count).

Area of a Sector (angle

θ

in degrees)

A = \frac{θ}{360} \cdot π r^{2}

Exterior Angle Theorem for Triangles

An exterior angle of a triangle equals the sum of the two opposite remote interior angles.

Semicolon Usage in ACT English

A semicolon joins two independent clauses without a coordinating conjunction: Independent clause; independent clause.

Comma + FANBOYS Rule

Use a comma before a coordinating conjunction (for, and, nor, but, or, yet, so) only when connecting two independent clauses.

Apostrophe for Possessive Singular vs. Plural

Singular: the student's book. Plural ending in s: the students' books. Irregular plural: the children's toys.

Colon Usage in ACT English

Must follow an independent complete sentence and introduce an explanation, list, or quote: She had one goal: to win.

Dangling Modifiers Correction

An introductory descriptive modifier must be immediately followed by the specific noun it describes.

Who vs. Whom

Who functions as the subject (replaces he/she); whom functions as the object of a verb or preposition (replaces him/her).

Its vs. It's vs. Its'

It's = contraction for it is. Its = possessive pronoun. Its' does NOT exist in standard English.

Redundancy and Conciseness in ACT English

Choose the shortest grammatically correct option that eliminates repetitive wording (e.g. avoid 'annual yearly event').

Surface Area of a Cylinder

A = 2 π r^{2} + 2 π r h

Volume of a Rectangular Prism

V = l \cdot w \cdot h

Perpendicular Lines Slopes

Perpendicular lines have negative reciprocal slopes: $

m_{1} \cdot m_{2} = - 1 ⟹ m_{2} = - \frac{1}{m _{1}}

Midpoint Formula in 3D Space

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})

Factorial Definition

n! = n \times (n - 1) \times \dots \times 1, 0! = 1

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