Precalculus Reference Sheet: Key Formulas and Identities

Factoring

Common Factoring Formulas

Factoring is a fundamental algebraic skill used to simplify expressions and solve equations. The following are standard factoring identities:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

• Sum of Squares: $a^2 + b^2$ is prime (cannot be factored over the real numbers)

• Perfect Square Trinomials:

  • $a^2 + 2ab + b^2 = (a + b)^2$

  • $a^2 - 2ab + b^2 = (a - b)^2$

• Sum and Difference of Cubes:

  • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

  • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Example:

Factor $x^2 - 9$:

$x^2 - 9 = (x - 3)(x + 3)$

Analytic Geometry

Basic Formulas

Analytic geometry connects algebra and geometry through coordinate systems.

• Slope of a Line: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form of a Line: $y - y_1 = m(x - x_1)$

• Distance Between Two Points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example:

Find the distance between $(1,2)$ and $(4,6)$:

$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Exponent Rules

Properties of Exponents

Exponents follow specific rules that simplify expressions and solve equations.

• $a^{x+y} = a^x a^y$

• $(ab)^x = a^x b^x$

• $(a^x)^y = a^{xy}$

• $a^0 = 1$ if $a \neq 0$

• $a^{-x} = \frac{1}{a^x}$ if $a \neq 0$

• $a^{x-y} = \frac{a^x}{a^y}$ if $a \neq 0$

Example:

Simplify $2^3 \cdot 2^4$:

$2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

Logarithm Rules

Properties of Logarithms

Logarithms are the inverses of exponentials and follow these rules:

• $\log_b x = y \iff x = b^y$

• $b^{\log_b x} = x$

• $\log_b b^x = x$

• $\log_b 1 = 0$

• $\log_b b = 1$

• $\log_b(xy) = \log_b x + \log_b y$

• $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

• $\log_b(x^y) = y \log_b x$

• $\log_b x = \frac{\log_a x}{\log_a b}$ (Change of base formula)

Example:

Evaluate $\log_2 8$:

$\log_2 8 = 3$ because $2^3 = 8$

Arithmetic Series

Formulas for Arithmetic Sequences and Series

An arithmetic sequence has a constant difference between terms. The following formulas are used:

• General Term: $a_k = a + (k-1)d$

• Sum of First n Terms: $S_n = \sum_{k=1}^n [a + (k-1)d] = \frac{n}{2}[2a + (n-1)d]$

• Alternatively: $S_n = \frac{n}{2}(a + a_n)$, where $a_n$ is the nth term

Example:

Find the sum of the first 5 terms of $2, 5, 8, ...$:

$a = 2$, $d = 3$, $n = 5$

$S_5 = \frac{5}{2}[2 \cdot 2 + (5-1) \cdot 3] = \frac{5}{2}[4 + 12] = \frac{5}{2} \cdot 16 = 40$

Geometric Series

Formulas for Geometric Sequences and Series

A geometric sequence has a constant ratio between terms. The following formulas are used:

• General Term: $a_n = a r^{n-1}$

• Sum of First n Terms: $S_n = \sum_{k=0}^{n-1} a r^k = a \frac{1 - r^n}{1 - r}$, $r \neq 1$

• Sum of Infinite Series: $S = \sum_{k=0}^{\infty} a r^k = \frac{a}{1 - r}$, $|r| < 1$

Example:

Find the sum of the first 4 terms of $3, 6, 12, ...$:

$a = 3$, $r = 2$, $n = 4$

$S_4 = 3 \frac{1 - 2^4}{1 - 2} = 3 \frac{1 - 16}{-1} = 3 \cdot 15 = 45$

Trigonometry

Basic Trigonometric Ratios

Trigonometric functions relate the angles of a right triangle to the ratios of its sides.

• $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\tan A = \frac{\text{opposite}}{\text{adjacent}}$

Special Angles Table

Values of sine, cosine, and tangent for common angles:

Angle	Radians	sin	cos	tan
0°	$0$	$0$	$1$	$0$
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	$1$	$0$	undefined

Pythagorean Identities

• $\cos^2 A + \sin^2 A = 1$

• $1 + \tan^2 A = \sec^2 A$

• $1 + \cot^2 A = \csc^2 A$

Ratio Identities

• $\tan A = \frac{\sin A}{\cos A}$

• $\cot A = \frac{\cos A}{\sin A}$

Reciprocal Identities

• $\sec A = \frac{1}{\cos A}$

• $\csc A = \frac{1}{\sin A}$

• $\cot A = \frac{1}{\tan A}$

Sum and Difference Identities

• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle Identities

• $\cos 2A = \cos^2 A - \sin^2 A$

• $\cos 2A = 2\cos^2 A - 1$

• $\cos 2A = 1 - 2\sin^2 A$

• $\sin 2A = 2\cos A \sin A$

• $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

Half Angle Identities

• $\cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}$

• $\sin \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{2}}$

• $\tan \frac{A}{2} = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A}$

Triple Angle Identities

• $\cos 3A = 4\cos^3 A - 3\cos A$

• $\sin 3A = 3\sin A - 4\sin^3 A$

Power Reduction Identities

• $\cos^2 A = \frac{1 + \cos 2A}{2}$

• $\sin^2 A = \frac{1 - \cos 2A}{2}$

• $\tan^2 A = \frac{1 - \cos 2A}{1 + \cos 2A}$

• $\cos^3 A = \frac{3\cos A + \cos 3A}{4}$

• $\sin^3 A = \frac{3\sin A - \sin 3A}{4}$

Sum-to-Product Identities

• $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$

• $\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

• $\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$

• $\cos A - \cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

Product-to-Sum Identities

• $\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$

• $\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$

• $\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

Sums of Sines and Cosines

• $A \cos x + B \sin x = \sqrt{A^2 + B^2} \sin(x + \varphi)$, where $\cos \varphi = \frac{B}{\sqrt{A^2 + B^2}}$, $\sin \varphi = \frac{A}{\sqrt{A^2 + B^2}}$

• $A \cos x + B \sin x = \sqrt{A^2 + B^2} \cos(x - \varphi)$, where $\cos \varphi = \frac{A}{\sqrt{A^2 + B^2}}$, $\sin \varphi = \frac{B}{\sqrt{A^2 + B^2}}$

Laws of Sines and Cosines

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Area of a Triangle

• Heron's Formula: $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$, where $s = \frac{a + b + c}{2}$

• Using Sine: $\text{Area} = \frac{1}{2}ab \sin C$

• Alternative Formula: $\text{Area} = \frac{c^2 \sin A \sin B}{2 \sin C}$

Circular Section

• Arc Length: $s = r\theta$ (where $\theta$ is in radians)

• Area of Sector: $A = \frac{1}{2} r^2 \theta$