Algebra 2 · formula sheet

Algebra 2 and trigonometry identities

Exponent and logarithm rules, complex numbers, sequences and series, the trigonometric identities, exact values, and the laws of sines and cosines.

19 formulas and facts in 6 sections

Exponents and logarithms

Definition: $\log_b x = y \iff b^{y} = x \qquad (b > 0,\; b \neq 1,\; x > 0)$
Product, quotient and power rules: $\log_b(xy) = \log_b x + \log_b y \qquad \log_b\frac{x}{y} = \log_b x - \log_b y \qquad \log_b x^{k} = k\log_b x$
Change of base: $\log_b x = \frac{\ln x}{\ln b}$
Inverse relationships: $b^{\log_b x} = x \qquad \log_b b^{x} = x$

Imaginary unit: $i^{2} = -1$
Conjugate product and modulus: $(a+bi)(a-bi) = a^{2} + b^{2} \qquad |a+bi| = \sqrt{a^{2}+b^{2}}$

Arithmetic: $a_n = a_1 + (n-1)d \qquad S_n = \frac{n}{2}(a_1 + a_n)$
Geometric: $a_n = a_1 r^{\,n-1} \qquad S_n = \frac{a_1(1-r^{n})}{1-r} \;(r \neq 1)$
Infinite geometric series: $S_{\infty} = \frac{a_1}{1-r} \quad\text{for } |r| < 1$
Binomial theorem: $(a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{k} \qquad \binom{n}{k} = \frac{n!}{k!\,(n-k)!}$

Pythagorean identities: $\sin^{2}\theta + \cos^{2}\theta = 1 \qquad 1 + \tan^{2}\theta = \sec^{2}\theta \qquad 1 + \cot^{2}\theta = \csc^{2}\theta$
Sum and difference: $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B \qquad \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
Double angle: $\sin 2A = 2\sin A\cos A \qquad \cos 2A = \cos^{2}A - \sin^{2}A = 2\cos^{2}A - 1 = 1 - 2\sin^{2}A$
Double angle for tangent: $\tan 2A = \frac{2\tan A}{1 - \tan^{2}A}$

Sine and cosine at the standard angles: $\begin{array}{c|ccccc} \theta & 0 & \tfrac{\pi}{6} & \tfrac{\pi}{4} & \tfrac{\pi}{3} & \tfrac{\pi}{2} \\ \hline \sin\theta & 0 & \tfrac{1}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{3}}{2} & 1 \\ \cos\theta & 1 & \tfrac{\sqrt{3}}{2} & \tfrac{\sqrt{2}}{2} & \tfrac{1}{2} & 0 \end{array}$
Degrees and radians: $\pi \text{ radians} = 180^{\circ}$

Law of sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Law of cosines: $c^{2} = a^{2} + b^{2} - 2ab\cos C$
Area from two sides and the included angle: $\text{Area} = \frac{1}{2}ab\sin C$

A reference for revision. In an exam, use the formula sheet your exam board provides and the notation your course uses.