MI IV — Equation Bank

In words. In a right triangle, sine, cosine, and tangent of an acute angle equal opposite-over-hypotenuse, adjacent-over-hypotenuse, and opposite-over-adjacent respectively.

Domain / range. Defined for acute angles in a right triangle: 0 < θ < π/2.

In words. For a point (x, y) on the terminal side of angle θ at distance r from the origin, sine is y/r, cosine is x/r, and tangent is y/x.

Domain / range. θ is any real number (extends right-triangle trig to all angles, including obtuse and negative).

In words. On the unit circle, the point at angle θ from the positive x-axis has coordinates (cos θ, sin θ).

Domain / range. θ is any real number; coordinates always satisfy x² + y² = 1.

In words. Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively.

Domain / range. csc undefined where sin = 0; sec undefined where cos = 0; cot undefined where sin = 0.

In words. Tangent equals sine over cosine; cotangent equals cosine over sine.

Domain / range. tan undefined where cos = 0 (i.e., θ = π/2 + πk). cot undefined where sin = 0 (i.e., θ = πk).

In words. Sine and cosine of 30°, 45°, and 60° in their exact radical form. Tangent is the corresponding ratio.

Domain / range. Reference angles 30°, 45°, 60° in the first quadrant; extend to other quadrants using reference-angle rules and quadrant sign chart.

In words. The absolute value of any trig function at angle θ equals the same trig function evaluated at its reference angle; the sign comes from the quadrant.

Domain / range. Reference angle θ_ref is always in [0, π/2] (or [0°, 90°]). Quadrant sign chart: Q1 all positive; Q2 only sin/csc positive; Q3 only tan/cot positive; Q4 only cos/sec positive ("All Students Take Calculus").

In words. The area of any triangle equals one-half the product of two sides times the sine of the included angle, and this can be written three equivalent ways depending on which pair of sides plus included angle you pick.

Domain / range. C must be an interior angle of a triangle: 0 < C < π.

In words. The area of a parallelogram with adjacent sides a and b and included angle θ is the product of the sides times the sine of the included angle.

Domain / range. θ ∈ (0, π) is the interior angle of the parallelogram.

In words. In any triangle, the ratio of the sine of an angle to the length of the opposite side is constant. Equivalently, the ratio of a side to the sine of its opposite angle is constant.

Domain / range. Each angle is in (0, π); each side is positive. The three angles must sum to π.

In words. Given angle A and sides a (opposite A) and c, the critical height is h = c sin A. If a < h, no triangle exists. If a = h, one right triangle. If h < a < c, two triangles. If a ≥ c, exactly one triangle.

Domain / range. A is an acute interior angle (the ambiguity table assumes A is acute; if A is right or obtuse, at most one triangle exists).

In words. Each side of a triangle must be strictly less than the sum of the other two.

Domain / range. All three sides are positive real numbers. Violation of any one inequality means no triangle exists.

In words. The square of any side of a triangle equals the sum of squares of the other two sides minus twice their product times the cosine of the included angle. Pattern repeats: the angle in the cosine is opposite the side on the left.

Domain / range. C ∈ (0, π); when C = π/2, cos C = 0 and the formula reduces to the Pythagorean theorem.

In words. Rearranged Law of Cosines: the cosine of any angle equals the sum of squares of the two adjacent sides minus the square of the opposite side, all divided by twice the product of the adjacent sides.

Domain / range. Output cos C ∈ [-1, 1]; if computed value exceeds 1 in absolute value, the three given sides cannot form a triangle.

In words. An angle bisector from a vertex divides the opposite side into two segments whose ratio equals the ratio of the two adjacent sides.

Domain / range. Applies to any triangle with a bisector drawn from one vertex to the opposite side.

In words. The inverse sine function maps a value in [-1, 1] to the unique angle in [-π/2, π/2] whose sine equals that value.

Domain / range. Domain: [-1, 1]. Range: [-π/2, π/2]. Equivalent in degrees: range [-90°, 90°].

In words. The inverse cosine function maps a value in [-1, 1] to the unique angle in [0, π] whose cosine equals that value.

Domain / range. Domain: [-1, 1]. Range: [0, π]. Equivalent in degrees: range [0°, 180°].

In words. The inverse tangent function maps any real number to the unique angle in the open interval (-π/2, π/2) whose tangent equals that value.

Domain / range. Domain: all real numbers (-∞, ∞). Range: open interval (-π/2, π/2). Equivalent in degrees: open range (-90°, 90°).

In words. The matrix that rotates any point in the plane counter-clockwise by angle θ about the origin has cosine on the diagonal, -sine in the top-right entry, and +sine in the bottom-left entry.

Domain / range. θ is any real number (in radians, conventionally). Determinant = cos²θ + sin²θ = 1, so rotations preserve distances and areas.

In words. The 2x2 identity matrix has ones on the main diagonal and zeros elsewhere; multiplying any vector by I leaves it unchanged.

Domain / range. Applies to any 2D vector (or 2x2 matrix). I is the zero-rotation case: $R_0$ = I.

In words. Rotating by β and then by α gives a single rotation by the sum α + β. This is the parent identity from which all sum-of-angles formulas descend.

Domain / range. α, β are any real numbers (in radians).

In words. The sum of the squares of sine and cosine of any angle equals one.

Domain / range. θ is any real number.

In words. Tangent squared plus one equals secant squared.

Domain / range. Undefined where cos θ = 0, i.e., θ = π/2 + πk.

In words. One plus cotangent squared equals cosecant squared.

Domain / range. Undefined where sin θ = 0, i.e., θ = πk.

In words. Cosine of the sum of two angles equals the product of cosines minus the product of sines.

Domain / range. α, β are any real numbers.

In words. Sine of the sum of two angles equals sine-times-cosine plus cosine-times-sine (mixed cross-products, added).

Domain / range. α, β are any real numbers.

In words. Cosine of the difference of two angles equals the product of cosines plus the product of sines (note the sign flip from the sum formula).

Domain / range. α, β are any real numbers.

In words. Sine of the difference of two angles equals sine-times-cosine minus cosine-times-sine.

Domain / range. α, β are any real numbers.

In words. Sine is odd (sign flips under negation of input). Cosine is even (preserved under negation). Tangent is odd (a ratio of an odd over an even is odd).

Domain / range. θ is any real number for sine and cosine; tangent excludes π/2 + πk.

In words. Sine and cosine swap when the input is replaced by its complement (π/2 minus the angle). Similarly for tan/cot and sec/csc — each function equals its co-function applied to the complementary angle.

Domain / range. α is any real number (for sine/cosine); tan/cot version excludes the singularities of the relevant function.

In words. Shifting the input by π or 3π/2 produces predictable sign flips or function swaps. These are special cases of the sum-of-angles identities.

Domain / range. α, x are any real numbers.

In words. Tangent of the sum of two angles equals the sum of tangents over one minus the product of tangents.

Domain / range. Undefined when tan α tan β = 1 (i.e., when α + β = π/2 + πk).

In words. Tangent of the difference of two angles equals the difference of tangents over one plus the product of tangents.

Domain / range. Undefined when tan α tan β = -1 (i.e., when α - β = π/2 + πk).

In words. The slope of a line equals the tangent of its angle of inclination (the angle measured counter-clockwise from the positive x-axis to the line, restricted to [0°, 180°)).

Domain / range. α ∈ [0, π) by convention (a line and its reversal have the same inclination). For vertical lines, α = π/2 and the slope is undefined.

In words. The tangent of the acute angle between two lines with slopes m₁ and m₂ equals the absolute value of their slope difference divided by one plus their slope product.

Domain / range. Defined whenever 1 + m₁m₂ ≠ 0 (i.e., when the lines are NOT perpendicular). If 1 + m₁m₂ = 0, then m₁m₂ = -1 and the lines are perpendicular (angle = 90°).

In words. Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. This is the denominator-zero case of the angle-between-lines formula.

Domain / range. Both lines must be non-vertical (otherwise slopes don't exist; perpendicularity is handled by inspection — horizontal and vertical lines are perpendicular).

In words. Sine of double an angle equals twice the product of sine and cosine of that angle.

Domain / range. α is any real number.

In words. Cosine of double an angle has three equivalent forms: cosine squared minus sine squared, or twice cosine squared minus one, or one minus twice sine squared.

Domain / range. α is any real number.

In words. Tangent of double an angle equals twice the tangent over one minus tangent squared.

Domain / range. Undefined when tan²α = 1 (i.e., α = π/4 + π/2 · k).

In words. Cosine squared of an angle equals one-half plus half the cosine of double the angle. Used to reduce powers of cosine to first-order expressions in cosine of a doubled angle.

Domain / range. α is any real number.

In words. Sine squared of an angle equals one-half minus half the cosine of double the angle.

Domain / range. α is any real number.

In words. Tangent squared of an angle equals one minus cosine of the doubled angle, divided by one plus cosine of the doubled angle.

Domain / range. Undefined when 1 + cos(2α) = 0, i.e., when 2α = π + 2πk (α = π/2 + πk — where tangent itself is undefined).

In words. Sine of triple an angle equals three times sine minus four times sine cubed.

Domain / range. x is any real number.

In words. Sine of double an angle can be written as twice the tangent divided by one plus tangent squared.

Domain / range. Undefined where tan θ is undefined (θ = π/2 + πk). Useful when only tangent of the angle is known.

In words. The sum of tangent and cotangent of an angle equals twice the cosecant of the doubled angle.

Domain / range. θ ≠ πk/2 (excludes points where sin or cos is zero).

In words. All solutions of sin θ = c come in two families: the principal value plus full revolutions, and the supplement of the principal value plus full revolutions.

Domain / range. c ∈ [-1, 1]; k ∈ ℤ. If |c| > 1, no solutions.

In words. All solutions of cos θ = c are the principal value (±) plus full revolutions.

Domain / range. c ∈ [-1, 1]; k ∈ ℤ. If |c| > 1, no solutions.

In words. All solutions of tan θ = c are the principal value plus half-revolutions (period of tangent is π, not 2π).

Domain / range. c is any real number; k ∈ ℤ.

In words. Two sines are equal exactly when the angles differ by a full revolution or are supplementary (mod 2π).

Domain / range. α, β are any real numbers; k ∈ ℤ.

In words. Any linear combination of sine and cosine of the same argument can be rewritten as a single sinusoid with amplitude √(A² + B²) and phase shift φ.

Domain / range. x is any real number. A and B are any real numbers (not both zero).

In words. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

Domain / range. Any real angle measure. Mental anchors: 180° = π rad, 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6.

In words. An angle in degrees, minutes, seconds is converted to decimal degrees by adding the degrees, the minutes divided by 60, and the seconds divided by 3600.

Domain / range. x integer (degrees), 0 ≤ y < 60 (minutes), 0 ≤ z < 60 (seconds). 1° = 60' = 3600''.

In words. A 2D vector v has two equivalent notations: angle-bracket components, basis-vector form using i and j, or a column matrix. The first component is the x-component (how far east); the second is the y-component (how far north).

In words. The magnitude (length) of vector ⟨a, b⟩ equals the square root of the sum of squared components. It is the Pythagorean distance from the tail to the tip.

Domain / range. Always non-negative; equals 0 iff v is the zero vector.

In words. For vector ⟨a, b⟩, the angle theta measured counterclockwise from the positive x-axis has tangent equal to b/a — BUT you must sketch first and add 180° if the vector lies in Q2 or Q3, because arctan only outputs angles in (−90°, 90°).

Domain / range. Defined for nonzero vectors; result in [0°, 360°) or (−180°, 180°] depending on convention.

In words. A vector with magnitude r at angle theta (measured counterclockwise from the positive x-axis) has rectangular components ⟨r·cos(theta), r·sin(theta)⟩. Magnitude times direction.

Domain / range. r >= 0; theta any real angle (typically [0°, 360°) or [0, 2π)).

In words. For a vector with magnitude r and bearing B (measured clockwise from due North), the x-component (east) is r·sin(B) and the y-component (north) is r·cos(B). Sine and cosine swap roles relative to the standard math convention.

Domain / range. r >= 0; B any real angle, typically [0°, 360°).

In words. A vector of magnitude r at compass direction 'N alpha degrees E' means: start pointing North, rotate alpha degrees toward East. The x-component is +r·sin(alpha); the y-component is +r·cos(alpha). Both positive because the vector lies in the NE quadrant (Q1).

Domain / range. r >= 0; alpha in [0°, 90°].

In words. A vector of magnitude r at compass direction 'N alpha degrees W' means: start pointing North, rotate alpha degrees toward West. The x-component is −r·sin(alpha) (negative because west); the y-component is +r·cos(alpha) (positive because north).

Domain / range. r >= 0; alpha in [0°, 90°].

In words. A vector of magnitude r at compass direction 'S alpha degrees E' means: start pointing South, rotate alpha degrees toward East. The x-component is +r·sin(alpha) (positive — east); the y-component is −r·cos(alpha) (negative — south).

Domain / range. r >= 0; alpha in [0°, 90°].

In words. A vector of magnitude r at compass direction 'S alpha degrees W' means: start pointing South, rotate alpha degrees toward West. Both components are negative — the vector lies in Q3.

Domain / range. r >= 0; alpha in [0°, 90°].

In words. The math angle theta (counterclockwise from +x) and the bearing B (clockwise from +y) for the same vector are complementary in a flipped sense: theta = 90° − B.

Domain / range. Any angle pair.

In words. To add two vectors, add their components separately. Geometrically: place v2's tail at v1's tip; the sum is the arrow from v1's tail to v2's tip (tip-to-tail). Equivalent: build a parallelogram with v1 and v2 as adjacent sides; the diagonal from the shared corner is the sum.

Domain / range. Always defined; commutative and associative.

In words. To subtract vectors, subtract their components. Equivalent: add v to the opposite of w. Geometrically: v − w points from the tip of w to the tip of v.

Domain / range. Always defined.

In words. The opposite of v has the same magnitude as v but points in the reverse direction. Negate every component.

Domain / range. Always defined.

In words. The magnitude of v − w is the distance between the tips of v and w (when both share a tail). You must subtract the vectors FIRST, then take the magnitude; |v − w| is generally NOT equal to |v| − |w|.

Domain / range. Always non-negative.

In words. The actual velocity of a vehicle relative to the ground equals its velocity relative to the air/water (heading) PLUS the air/water's velocity relative to the ground (wind or current). Solve for $v_h$eading when you know the desired ground velocity: $v_h$eading = $v_g$round − $v_w$ind.

Domain / range. All velocity vectors in the same frame.

In words. Multiplying a vector by a scalar c scales every component by c. The magnitude scales by |c|; if c > 0 the direction is preserved; if c < 0 the direction is reversed; if c = 0 you get the zero vector.

Domain / range. c any real scalar; v any vector.

In words. A linear combination of vectors x and y with scalar coefficients a and b. Any 2D vector v can be written uniquely as a linear combination of two non-parallel vectors x and y — they form a basis for the plane. Solve by equating components to get a system of two equations in two unknowns.

Domain / range. Requires x and y to be linearly independent (not parallel) for a unique solution.

In words. In a triangle, the segment joining the midpoints of two sides is parallel to the third side and exactly half its length. Vector proof reduces to a one-line calculation: MN = (1/2)·BC.

Domain / range. Any triangle.

In words. In parallelogram ABCD, the diagonals AC and BD cross at a point E that is the midpoint of BOTH diagonals. Equivalently: AE = (1/2)·AC and BE = (1/2)·BD.

Domain / range. Any parallelogram.

In words. The unit vector v-hat in the direction of v is v divided by its own magnitude. The result has length 1 and points the same way as v. The 'pure direction' of v with magnitude stripped out.

Domain / range. Defined for any nonzero vector v.

In words. Every vector splits as the product of its magnitude (how long) and its unit vector (which way). This is THE foundational identity for the entire unit — every other formula sits on top of it.

Domain / range. Any nonzero vector.

In words. The unit vector at angle theta (measured CCW from the positive x-axis) has components ⟨cos(theta), sin(theta)⟩. This is the bridge between trigonometry (angles) and vectors (direction): every unit vector in 2D is a point on the unit circle.

Domain / range. theta any real angle.

In words. The two standard unit vectors. i points east (along +x), j points north (along +y). Every 2D vector ⟨a, b⟩ = a·i + b·j.

In words. The dot product of two vectors is the sum of products of corresponding components. It returns a scalar (single number), not a vector. Extends to 3D by adding the third component product.

Domain / range. Defined for two vectors of the same dimension; output is a real scalar.

In words. The dot product also equals the product of the magnitudes times the cosine of the angle between the vectors. Combined with the algebraic form, this lets you recover the angle from purely algebraic data (and vice versa).

Domain / range. theta in [0°, 180°]: the geometric angle between v and w when placed tail-to-tail.

In words. The cosine of the angle between two vectors equals their dot product divided by the product of their magnitudes. Take arccos to recover theta. Works in any dimension. Range of theta is [0°, 180°].

Domain / range. Defined for nonzero vectors; theta in [0°, 180°].

In words. Two nonzero vectors are perpendicular if and only if their dot product is zero. This is the single most useful application of the dot product: a one-line algebraic test for orthogonality.

Domain / range. Both vectors nonzero.

In words. You can classify the angle between two vectors without computing it: positive dot product means acute, zero means perpendicular, negative means obtuse. (Caveat: positive can also mean parallel/same direction, theta = 0°; negative can mean anti-parallel, theta = 180°. Always check parallelism before declaring 'acute' or 'obtuse.')

Domain / range. Both vectors nonzero.

In words. Two nonzero vectors are parallel if and only if one is a scalar multiple of the other. If k > 0, they point the same direction (theta = 0°); if k < 0, anti-parallel (theta = 180°). Equivalent component-wise test in 2D: $v_1$/$w_1$ = $v_2$/$w_2$ (proportional components).

Domain / range. Both vectors nonzero.

In words. The equation a·v = c (where v = ⟨x, y⟩ is variable and a = ⟨$a_1$, $a_2$⟩ is fixed) defines a LINE. The vector a is the NORMAL VECTOR — perpendicular to the line. Changing c shifts the line parallel to itself without changing its normal direction. Every linear equation ax + by = c hides this dot product structure: the coefficients form the normal vector.

Domain / range. a nonzero; c any real.

In words. The dot product is commutative (order doesn't matter), distributive over vector addition, and compatible with scalar multiplication (the scalar can move outside).

Domain / range. Any vectors and scalars.

In words. The dot product of a vector with itself equals the square of its magnitude. Convenient identity used inside the projection formula and elsewhere.

Domain / range. Any vector v.

In words. The projection of v onto w equals the scalar (|v|·cos(theta)) — which is the signed length of the shadow — multiplied by the unit vector w-hat — which gives it direction. 'Length-times-direction' decomposition of the projection.

Domain / range. w nonzero; theta = angle between v and w.

In words. The projection of v onto w equals a scalar coefficient (the ratio of two dot products) times the vector w itself. NO cosine, NO arccos, NO explicit magnitudes — just dot products. This is the workhorse formula. Same form in 2D and 3D.

Domain / range. w nonzero.

In words. The projection formula handles acute, right, and obtuse angles uniformly. The sign of v·w determines whether pro$j_w$ v points along w (positive, acute angle), is zero (perpendicular), or points opposite to w (negative, obtuse angle). The formula has no triangle in it — purely algebraic, valid in all three cases.

Domain / range. w nonzero; theta in [0°, 180°].

In words. The projection of v onto w is the zero vector if and only if v and w are perpendicular. The shadow of v has zero length because v is hitting w's line edge-on.

Domain / range. w nonzero.

In words. The projection of v onto w equals v itself if and only if v is parallel to w. The shadow IS the vector — because v already lies along w's line.

Domain / range. w nonzero.

In words. The 2x2 matrix that rotates any 2D vector counterclockwise by theta. Apply via matrix-vector multiplication: R(theta)·v gives v rotated by theta. The columns of R are the rotated basis vectors: first column = where i lands, second column = where j lands.

Domain / range. theta any real angle.

In words. Rotation matrices preserve magnitudes (isometries), preserve angles, have determinant 1 (area-preserving and orientation-preserving), and their inverse equals their transpose (which is the same as rotation by negative theta, i.e., clockwise rotation by the same angle).

Domain / range. Any theta.

In words. Composing two rotations is equivalent to a single rotation by the sum of the angles. The matrix product encodes the trig angle-sum identities: when you multiply out R(alpha)·R(beta), the resulting entries are cos(alpha+beta), −sin(alpha+beta), sin(alpha+beta), cos(alpha+beta).

Domain / range. Any angles alpha, beta. 2D rotations commute; 3D rotations in general do not.

In words. On a ramp inclined at angle theta above horizontal, the component of weight parallel to the ramp surface (the part that tries to slide the object down) equals weight times sin(theta).

Domain / range. theta in [0°, 90°].

In words. On a ramp at angle theta, the component of weight perpendicular to the ramp (pressing the object into the surface) equals weight times cos(theta).

Domain / range. theta in [0°, 90°].

In words. The ratio of tangential to normal force component equals the tangent of the ramp angle. Lets you find theta without knowing the actual weight — the weight cancels.

Domain / range. theta in [0°, 90°); $w_N$ > 0.

In words. An object is in equilibrium (not accelerating) when the vector sum of all forces on it equals the zero vector. In 2D this single vector equation becomes TWO scalar equations: sum of x-components = 0, sum of y-components = 0. With two unknowns, the system is solvable.

Domain / range. Any system of force vectors.

In words. An object of mass m has weight vector pointing straight down (negative y) with magnitude m·g, where g ≈ 9.8 m/s² is gravitational acceleration. Units: kilograms times m/s² = Newtons.

Domain / range. m > 0; g ≈ 9.8 m/s² near Earth's surface.

In words. A cord tension with magnitude |T|, oriented at angle alpha measured CCW from the positive x-axis, has components |T|·cos(alpha) and |T|·sin(alpha). For a cord pulling up-left at angle beta above horizontal, alpha = 180° − beta. For up-right at angle beta, alpha = beta.

Domain / range. |T| >= 0; alpha any angle (typically chosen so the cord pulls 'up' away from the object).

In words. A 3D vector has three components corresponding to the three coordinate axes. The new basis vector k = ⟨0, 0, 1⟩ points in the +z direction (up). Same notation conventions as 2D, just one more component.

In words. The three standard 3D unit vectors: i along +x (east), j along +y (north), k along +z (up). Mutually perpendicular, each of length 1.

In words. The magnitude of a 3D vector is the square root of the sum of the three squared components. Pythagoras applied twice (stacked): once for the base diagonal in the xy-plane, again for the full 3D diagonal.

Domain / range. Always non-negative; equals 0 only for the zero vector.

In words. The 3D dot product is the sum of the three products of corresponding components. The geometric identity v·w = |v|·|w|·cos(theta) holds unchanged in 3D — same angle interpretation as 2D.

Domain / range. Both vectors 3D.

In words. Any 3D vector can be written uniquely as a linear combination of three linearly independent basis vectors x, y, z. Component-wise equality produces a 3x3 linear system in unknowns a, b, c.

Domain / range. Requires x, y, z linearly independent — equivalent to det[x | y | z] ≠ 0.

In words. The angles a 3D vector makes with each coordinate axis have cosines equal to the corresponding components of the unit vector. The squared direction cosines sum to 1 — this is just the magnitude-of-unit-vector identity in disguise.

Domain / range. v nonzero; alpha, beta, gamma in [0°, 180°].

In words. The cross product of two 3D vectors is computed as the symbolic determinant of a 3x3 matrix with the basis vectors i, j, k in the first row and the components of v and w in the second and third rows. Expand along the first row with sign pattern +, −, +.

Domain / range. Defined only in 3D. Output is a 3D vector.

In words. Direct component formula for the cross product. Each output component is a 2x2 determinant of the 'other two columns' from the input vectors. Equivalent to the determinant form, just written out.

Domain / range. v, w in 3D.

In words. The magnitude of the cross product equals the product of the input magnitudes times the sine of the angle between them — which is exactly the area of the parallelogram with sides v and w. This is the geometric companion to the dot product (which extracts cosine).

Domain / range. theta in [0°, 180°]; the magnitude is always non-negative.

In words. Point your right hand's fingers in the direction of v. Curl them toward w (through the shorter angle). Your thumb points in the direction of v × w. The result is always perpendicular to both v and w; the right-hand rule disambiguates which of the two perpendicular directions you get.

Domain / range. Defined whenever v × w is nonzero.

In words. Swapping the order of the cross product flips the result. The right-hand rule's thumb points the opposite way when you curl from w to v instead of v to w. This is the defining feature that distinguishes cross product from addition, multiplication, and dot product (all of which are commutative).

Domain / range. Any 3D v, w.

In words. By construction the cross product v × w is perpendicular to BOTH v and w. Dotting the cross product with either input always gives zero. Useful as a self-check after computing a cross product.

Domain / range. Any 3D v, w.

In words. Two nonzero 3D vectors are parallel if and only if their cross product is the zero vector. The 'parallel detector' in 3D — analog of the dot product's perpendicularity detector. Also: v × v = 0 always.

Domain / range. Both vectors 3D.

In words. The area of the triangle formed by two vectors v and w (originating at a common vertex) is half the magnitude of their cross product. A triangle is exactly half of the parallelogram spanned by its two sides.

Domain / range. v, w nonzero and not parallel.

In words. To find the angle between two vectors, ALWAYS use the dot product formula with arccos. The cross-product formula sin(theta) = |v×w|/(|v|·|w|) seems analogous, but arcsin only outputs angles in [−90°, 90°] and cannot distinguish acute from obtuse — you can get an answer 180° off the truth.

Domain / range. theta in [0°, 180°] for the angle between vectors.

In words. The scalar triple product takes three 3D vectors and produces a scalar equal to the determinant of the 3x3 matrix with the vectors as rows. Its absolute value equals the volume of the parallelepiped spanned by the three vectors.

Domain / range. u, v, w in 3D.

In words. The volume of the parallelepiped with edges u, v, w is the absolute value of the scalar triple product.

Domain / range. u, v, w in 3D, linearly independent (else volume = 0).

In words. The volume of a tetrahedron with three edge vectors u, v, w meeting at one vertex equals one-sixth of the absolute value of the scalar triple product. A tetrahedron is 1/6 of the corresponding parallelepiped.

Domain / range. u, v, w in 3D, linearly independent.

In words. Given three vertices A, B, C of a parallelogram, the fourth vertex D is found using the fact that the diagonals bisect each other. If A and C are opposite (diagonal AC), then midpoint of AC equals midpoint of BD, so D = A + C − B. Three different pairings of 'diagonal' give three possible fourth vertices.

Domain / range. Any three points in 2D or 3D not all collinear.

In words. Dot and cross product are the geometric companions of cosine and sine. Dot extracts the cosine (and is scalar, defined in all dimensions, commutative, zero iff perpendicular). Cross extracts the sine and a perpendicular direction (and is a 3D-only vector, anti-commutative, zero iff parallel). Together they characterize the entire geometric relationship between two vectors.

Domain / range. theta in [0°, 180°].

Domain / range. Roots of a$x^2$ + bx + c = 0 with a, b, c real (or complex) and a $\neq$ 0.

Domain / range. Real coefficients a, b, c with a $\neq$ 0.

Domain / range. z on the unit circle; n a positive integer.

Domain / range. Polynomials of positive degree n over $\mathbb{C}$.

Domain / range. Positive integer n.

Domain / range. Foundational identity of $\mathbb{C}$.

Domain / range. Every z $\in$ $\mathbb{C}$.

Domain / range. All real $\theta$ (angles in degrees or radians).

Domain / range. Every z $\in$ $\mathbb{C}$; r $\in$ [0, $\infty$); $\theta$ $\in$ $\mathbb{R}$ (taken mod 360° or 2$\pi$).

Domain / range. Every z $\in$ $\mathbb{C}$; |z| $\ge$ 0 with equality iff z = 0.

Domain / range. z = a + bi $\neq$ 0. $\theta$ is defined modulo 360° (or 2$\pi$); principal value typically in [0°, 360°) or (-180°, 180°].

Domain / range. Nonzero z = a + bi with a, b $\in$ $\mathbb{R}$; $\alpha$ is the positive acute reference angle.

Domain / range. Every (r, $\theta$) with r $\ge$ 0; produces a unique z = a + bi.

Domain / range. Every z $\in$ $\mathbb{C}$.

Domain / range. Every z = a + bi $\in$ $\mathbb{C}$.

Domain / range. $\theta$ a rational multiple of 360° (e.g., an integer number of degrees). For irrational $\theta$ no finite n exists.

Domain / range. All real $\theta$.

Domain / range. All real $\alpha$, $\beta$.

Domain / range. All real $\alpha$, $\beta$.

Domain / range. All real $\alpha$, $\beta$.

Domain / range. All real $\alpha$, $\beta$.

Domain / range. All real $\alpha$, $\beta$.

Domain / range. Any $z_1$ = $r_1$$\,$cis$\,$$\theta$_1 and $z_2$ = $r_2$$\,$cis$\,$$\theta$_2 in $\mathbb{C}$; $r_1$, $r_2$ $\ge$ 0. Angles taken mod 360°.

Domain / range. $z_2$ $\neq$ 0 (i.e., $r_2$ > 0). Otherwise any $z_1$, $z_2$ $\in$ $\mathbb{C}$; angles mod 360°.

Domain / range. Every z $\in$ $\mathbb{C}$.

Domain / range. Integer exponents n (extends to negatives via $i^{-1}$ = -i).

Domain / range. Multiplication applied to any z $\in$ $\mathbb{C}$.

Domain / range. z $\in$ $\mathbb{C}$, w on the unit circle.

Domain / range. z, w $\in$ $\mathbb{C}$ with w = r$\,$cis($\alpha$), r $\ge$ 0.

Domain / range. Any expression where a negative scalar appears in front of a cis term.

Domain / range. Every z $\in$ $\mathbb{C}$.

Domain / range. Every z $\in$ $\mathbb{C}$.

Domain / range. z = r$\,$cis($\theta$) $\in$ $\mathbb{C}$, r $\ge$ 0, n a non-negative integer. Extends to n < 0 via division ($z^{-1}$ = $r^{-1}$$\,$cis(-$\theta$)) and to n = 0 trivially ($z^0$ = 1).

Domain / range. Nonzero z = r$\,$cis($\theta$), any integer n.

Domain / range. Non-negative integer n; equates the binomial expansion with DeMoivre's compact answer.

Domain / range. Nested products of cis-form expressions with integer exponents; particularly clean when all radii are 1.

Domain / range. Any z $\in$ $\mathbb{C}$, looking at the sequence $z^0$, $z^1$, $z^2$, $\dots$.

Domain / range. z = cis($\theta$) on the unit circle with rational angle (finite cycle); k any non-negative integer.

Domain / range. Positive integer n. Equation $z^n$ = 1 over $\mathbb{C}$ has exactly these n solutions.

Domain / range. Any nonzero complex R$\,$cis($\alpha$); positive integer n. Produces n distinct n-th roots.

In words. {$a_k$} from k=1 to n is the ordered list $a_1$, $a_2$, ..., $a_n$

In words. Sequence = ordered list. Series = sum of that list.

In words. Explicit: plug n directly. Recursive: define $a_n$ from previous term(s), need a starting value.

In words. Whenever you write a closed-form formula, plug n=1 first.

In words. Constant second difference c → quadratic with leading coefficient c/2

In words. Run differences AND ratios. Constant first diff → arithmetic. Constant ratio → geometric. Constant reciprocal-diff → harmonic. Constant p-th diff → polynomial degree p.

In words. $a_n$ = $a_1$ + (n-1) · d

In words. $a_1$ given; $a_n$ = $a_{n-1}$ + d for n > 1

In words. $a_n$ = $a_1$ · r^(n-1)

In words. $a_1$ given; $a_n$ = r · $a_{n-1}$ for n > 1

In words. Reciprocals of a harmonic sequence form an arithmetic sequence.

In words. AM = (a + b) / 2

In words. AM = ($a_1$ + $a_2$ + ... + $a_n$) / n

In words. GM = √(ab)

In words. GM = ($a_1$ · $a_2$ · ... · $a_n$)^(1/n)

In words. HM = 2ab / (a + b)

In words. HM = n / (1/$a_1$ + 1/$a_2$ + ... + 1/$a_n$)

In words. QM = √((a² + b²)/2)

In words. QM = √((Σ$a_i$²)/n)

In words. HM ≤ GM ≤ AM ≤ QM

In words. Common difference d = (B - A) / (k + 1)

In words. Common ratio r = (B/A)^(1/(k+1))

In words. Flip A and B, insert k arithmetic means in the reciprocal sequence, flip back

In words. Apply function f repeatedly to the previous output.

In words. x* is a fixed point if plugging it into f returns x* itself

In words. For linear f(x) = mx + b: attracting if |m| < 1, repelling if |m| > 1

In words. Attracting if |f'(x*)| < 1, repelling if |f'(x*)| > 1

In words. For linear iteration $x_n$ = m·$x_{n-1}$ + b, fixed point L = b / (1 - m)

In words. To solve g(x) = 0: rewrite as x = f(x), iterate, watch for convergence

In words. Iconic fixed points worth recognizing

In words. Quadratic iterations can have stable cycles of various periods

In words. Modular iterations partition the modular group into orbits

In words. The iteration x → 1 + 1/x generates ratios of consecutive Fibonacci numbers and converges to φ

In words. Σ from k=1 to n of $a_k$ = $a_1$ + $a_2$ + ... + $a_n$

In words. Π from k=1 to n of $a_k$ = $a_1$ · $a_2$ · ... · $a_n$

In words. Union from k=1 to n of $A_k$

In words. Intersection from k=1 to n of $A_k$

In words. If left endpoint sequence converges to L but never reaches L, then L is NOT in the union

In words. The sequence $S_1$, $S_2$, $S_3$, ... where each $S_n$ is the partial sum up to term n

In words. Σ c·$a_k$ = c · Σ $a_k$

In words. Σ ($a_k$ + $b_k$) = Σ $a_k$ + Σ $b_k$

In words. Σ from m to n = Σ from 1 to n - Σ from 1 to m-1

In words. Number of terms = n - m + 1 (NOT n - m, NOT n)

In words. Σ_{k=1}^{n} c = n · c

In words. Σ_{k=1}^{n} k = n(n+1)/2

In words. Σ_{k=1}^{n} k² = n(n+1)(2n+1)/6

In words. Σ_{k=1}^{n} k³ = [n(n+1)/2]² = (Σk)²

In words. Sum of degree-p polynomial in k → degree-(p+1) polynomial in n

In words. Σ_{k=1}^{n} ($b_k$ - $b_{k+1}$) = $b_1$ - $b_{n+1}$

In words. Σ_{k=1}^{∞} ($b_k$ - $b_{k+1}$) = $b_1$ - lim $b_{n+1}$

In words. Tail term unbounded → telescoping series diverges

In words. Factor into ratio of factorials, then cancel

In words. $S_n$ = (n/2)(2a + (n-1)d)

In words. $S_n$ = (n/2)(a + ℓ) where ℓ is the last term

In words. $S_n$ = a(1 - $r^n$) / (1 - r), provided r ≠ 1

In words. $S_n$ = n · a when r = 1

In words. Infinite arithmetic series ALWAYS diverges, except when a = 0 AND d = 0

In words. S = a / (1 - r), provided |r| < 1

In words. $r^n$ behavior as n → ∞

In words. Equally-spaced parallel slices of a trapezoid (bases A, B) form arithmetic sequence

In words. Area of annulus with outer radius n+1, inner radius n is π(2n+1)

In words. Length-p repeating block: block digits over p nines

In words. Scale by 1$0^a$ and 10^(a+p), subtract to eliminate the repeating part

In words. Σ(2k)² factors out: 4·Σk²

In words. h = ab/(a+b) = HM/2

In words. $t_c$ombined = ($t_A$ · $t_B$) / ($t_A$ + $t_B$)

In words. Total = h(1 + ρ) / (1 - ρ)

In words. Compound interest: r = 1 + rate (growth) or r = 1 - rate (decay)

In words. Identify type → pick formula → plug in

In words. If $a_k$ is degree-p polynomial in k, then $S_n$ is degree-(p+1) polynomial in n

In words. The five sequence types