Registered Electrical Engineer (REE) Licensure Examination — Trigonometry Core Memory Sheet

Vocabulary practice flashcards covering angle units, trigonometric identities, triangle laws, shortcuts, spherical trigonometry, and examination tactics from the REE Licensure Examination Trigonometry Core Memory Sheet.

Degree (°)

A unit of circular measure where one full circle equals $360^\theta$, divided into $1^\theta = 60'$ (minutes) and $1' = 60''$ (seconds).

Radian (rad)

A circular measure unit where one full revolution equals $2\times\theta\,\text{rad}$, with $1\,\text{rad} = \frac{180^\theta}{\theta} \thickapprox 57.2958^\theta$.

Gradian (grad / gon)

A system of angular measure where a full circle equals $400\,\text{grad}$ and one right angle ($90^\theta$) equals $100\,\text{grad}$.

Mil (Artillery / NATO)

A military angular measurement unit where a complete circle equals $6400\,\text{mils}$ and a right angle ($90^\theta$) equals $1600\,\text{mils}$.

Arc Length and Sector Area Formulas

For radius $r$ and angle $\theta$ measured in radians, the arc length is $s = r\times\theta$ and the sector area is $A = \frac{1}{2}r^2\theta = \frac{1}{2}rs$.

Co-function Identities

Identities relating complementary angles ($A + B = 90^\theta$): \sin(\theta) = \timesos(90^\theta - \theta), \tan(\theta) = \timesot(90^\theta - \theta), and \sec(\theta) = \timessc(90^\theta - \theta).

ASTC Quadrant Sign Rule

Rule determining trigonometric signs: Quadrant I (All positive), Quadrant II ($\sin$, $\csc$ positive), Quadrant III ($\tan$, $\cot$ positive), and Quadrant IV ($\cos$, $\sec$ positive).

Trigonometric Parity (Negative Angle Rules)

Cosine and secant are even functions where \cos(-\theta) = \timesos(\theta) and \sec(-\theta) = \timesec(\theta); all other functions are odd, such as \sin(-\theta) = -\timesin(\theta) and \tan(-\theta) = -\timesan(\theta).

Pythagorean Triples

Integer right-triangle side combinations that speed up calculations: $(3, 4, 5)$, $(5, 12, 13)$, $(7, 24, 25)$, $(8, 15, 17)$, $(20, 21, 29)$, and their multiples (such as $6-8-10$ or $10-24-26$).

Pythagorean Identities

Three fundamental trigonometric relationships: \sin^2(\theta) + \timesos^2(\theta) = 1, 1 + \timesan^2(\theta) = \timesec^2(\theta), and 1 + \timesot^2(\theta) = \timessc^2(\theta).

Double-Angle Identities

Identities expanding twice an angle: \sin(2\theta) = 2\timesin(\theta)\timesos(\theta) = \frac{2\timesan(\theta)}{1 + \timesan^2(\theta)}, \cos(2\theta) = \timesos^2(\theta) - \timesin^2(\theta) = 2\timesos^2(\theta) - 1 = 1 - 2\timesin^2(\theta) = \frac{1 - \timesan^2(\theta)}{1 + \timesan^2(\theta)}, and \tan(2\theta) = \frac{2\timesan(\theta)}{1 - \timesan^2(\theta)}.

Power-Reduction Identities

Identities that rewrite squared trigonometric functions into first powers: \sin^2(\theta) = \frac{1 - \timesos(2\theta)}{2} and \cos^2(\theta) = \frac{1 + \timesos(2\theta)}{2}.

Product-to-Sum Identities

Formulas useful in AC power and modulation: \sin(A)\timesos(B) = \frac{1}{2}[\sin(A + B) + \timesin(A - B)], \cos(A)\timesos(B) = \frac{1}{2}[\cos(A - B) + \timesos(A + B)], and \sin(A)\timesin(B) = \frac{1}{2}[\cos(A - B) - \timesos(A + B)].

Law of Sines

Oblique triangle law stating $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$, where $R$ is the circumradius; used given ASA, AAS, or SSA.

Law of Cosines

Oblique triangle formula given SAS or SSS: c^2 = a^2 + b^2 - 2ab\timesos(C) or rearranged as $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.

Law of Tangents

Trigonometric relationship used for checking SAS solutions or direct exam recall: $\frac{a - b}{a + b} = \frac{\tan\left(\frac{1}{2}(A - B)\right)}{\tan\left(\frac{1}{2}(A + B)\right)}$.

Heron's Formula

Formula for the area of any triangle ($A_T$) given three sides: A_T = \timesqrt{s(s - a)(s - b)(s - c)}, where semi-perimeter $s = \frac{a + b + c}{2}$.

Harmonic Combination

Converts a linear sum a\timesin(\theta) + b\timesos(\theta) into a single sinusoid R\timesin(\theta + \alpha), with amplitude R = \timesqrt{a^2 + b^2} and phase angle $\tan(\alpha) = \frac{b}{a}$.

Two-Observation Elevation Shortcut

Formula to find tower height $h$ when moving distance $d$ closer changes elevation from $\alpha$ to $\beta$: h = \frac{d}{\cot(\alpha) - \timesot(\beta)} = \frac{d\timesin(\alpha)\timesin(\beta)}{\sin(\beta - \alpha)}.

Equal Subtended Angles Shortcut

Formula finding horizontal distance $x$ where lower segment $h_1$ and upper segment $h_2$ of a tower subtend equal angles: x = h_1\timesqrt{\frac{h_2 + h_1}{h_2 - h_1}}.

Bearing

An acute angle ($0^\theta - 90^\theta$) measured starting from North or South toward East or West (e.g., N $35^\theta$ E, S $20^\theta$ W).

Azimuth

A clockwise angle ($0^\theta - 360^\theta$) measured from North (or from South in geodetic surveying).

Euler's Formula

Identity expressing complex exponential form: e^{\pm j\theta} = \timesos(\theta) \pm j\timesin(\theta) = \timesext{cis}(\pm\theta).

De Moivre's Theorem

Theorem for raising complex numbers to powers: (r\,\timesext{cis}(\theta))^n = r^n\,\timesext{cis}(n\theta).

Hyperbolic Functions

Functions used in catenary cables and transmission lines: $\sinh(x) = \frac{e^x - e^{-x}}{2}$, $\cosh(x) = \frac{e^x + e^{-x}}{2}$, and $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$, which satisfy \cosh^2(x) - \timesinh^2(x) = 1.

Spherical Excess (E)

The difference $E = (A + B + C) - 180^\theta$ for a spherical triangle, where $180^\theta < A + B + C < 540^\theta$, used to calculate surface area on a sphere of radius $R$: $\text{Area} = \frac{\pi R^2 E}{180^\theta}$.

Nautical Mile (nmi)

A unit equal to $1'$ (one arc-minute, or $1/60^\theta$) of a great circle on Earth, approximately equivalent to $6080\,\text{ft} = 1852\,\text{m}$.

Arbitrary Angle Substitution

An REE multiple-choice technique where a non-special angle (such as $\theta = 23^\theta$ or $37^\theta$) is stored in calculator variable A or X to test and match algebraic trigonometric identity options.