Trigonometric Identities

• Basic trigonometric identities, quotient identities, reciprocal identities, Pythagorean identities, cofunction identities, and negative angle identities

Quotient Identities

• tan θ = sin θ / cos θ, where cos θ ≠ 0
• cot θ = cos θ / sin θ, where sin θ ≠ 0

Reciprocal Identities

• sin θ = 1 / csc θ, where csc θ ≠ 0; csc θ = 1 / sin θ, where sin θ ≠ 0.
• cos θ = 1 / sec θ, where sec θ ≠ 0; sec θ = 1 / cos θ, where cos θ ≠ 0.
• tan θ = 1 / cot θ, where cot θ ≠ 0; cot θ = 1 / tan θ, where tan θ ≠ 0.

Pythagorean Identities

• cos² θ + sin² θ = 1
• tan² θ + 1 = sec² θ
• cot² θ + 1 = csc² θ

Cofunction Identities

• sin (π/2 - θ) = cos θ
• cos (π/2 - θ) = sin θ
• tan (π/2 - θ) = cot θ

Negative Angle Identities

• sin (-θ) = -sin θ
• cos (-θ) = cos θ
• tan (-θ) = -tan θ
• Mnemonic SOH, CAH, TOA defines basic trigonometric ratios.
• All trigonometric functions are positive in the 1st quadrant.
• Sine and cosecant are positive in the 2nd quadrant.
• Tangent and cotangent are positive in the 3rd quadrant.
• Cosine and secant are positive in the 4th quadrant.

Sum Identities

• sin (A + B) = sin A cos B + cos A sin B
• cos (A + B) = cos A cos B - sin A sin B
• tan (A + B) = (tan A + tan B) / (1 - tan A tan B)

Double-Angle Identities

• cos 2θ = cos² θ - sin² θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = 2 cos² θ - 1
• tan 2θ = (2 tan θ) / (1 - tan² θ)
• cos 2θ = 1 - 2 sin² θ

Half-Angle Identities

• sin (θ/2) = ±√((1 - cos θ) / 2)
• cos (θ/2) = ±√((1 + cos θ) / 2)
• tan (θ/2) = ±√((1 - cos θ) / (1 + cos θ)), where cos θ ≠ -1

Linear Systems

• To solve a system of n linear equations with n variables written as AX = B, calculate X = A⁻¹B.

Parabolas

• Vertical parabola equation: y = a(x - h)² + k.
  • Horizontal parabola equation: x = a(y - k)² + h.
• For a vertical parabola, if a > 0, opens upward; if a < 0, opens downward.
  • For a horizontal parabola, if a > 0, opens to the right; if a < 0, opens to the left.
• Vertex of both vertical and horizontal parabolas: (h, k).
• Axis of symmetry for vertical parabola: x = h
  • Axis of symmetry for horizontal parabola: y = k.
• Focus for a vertical parabola: (h, k + 1/(4a))
  • Focus for a horizontal parabola: (h + 1/(4a), k)
• Directrix for a vertical parabola: y = k - 1/(4a)
  • Directrix for a horizontal parabola: x = h - 1/(4a)
• Length of latus rectum for both parabolas: |1/a| units.

Equations of Circles

• Standard form: x² + y² = r²; center at (0, 0).
  • General form: (x - h)² + (y - k)² = r²; center at (h, k).
• In both forms, the radius is 'r'.

Ellipses Centered at (h, k)

• Standard form for horizontal ellipse: ((x-h)² / a²) + ((y-k)² / b²) = 1.
  • Standard form for vertical ellipse: ((y-k)² / a²) + ((x-h)² / b²) = 1.
• Orientation: horizontal or vertical.
• Foci equation: c² = a² - b²
• Major axis = 2a, Minor axis = 2b

Hyperbolas Centered at (h, k)

• Standard form for horizontal hyperbola: ((x-h)² / a²) - ((y-k)² / b²) = 1.
  • Standard form for vertical hyperbola: ((y-k)² / a²) - ((x-h)² / b²) = 1.
• Foci equation : c² = a² + b²
• Transverse axis = 2a, Conjugate axis = 2b

Systems of Equations

• Systems of equations consisting of linear and nonlinear equations can have zero, one, or two solutions.
• Quadratic systems with conic sections can have anywhere from zero to four solutions.
• Linear-quadratic systems can be solved using graphical or algebraic methods.
• Quadratic-quadratic systems can be solved using elimination.

Parametric Equations

• Representing relationships between variables where x and y are expressed in terms of a third variable.

Vectors

• Vectors are represented geometrically by directed line segments indicating magnitude and direction.
• A vector has an initial point (tail) and a terminal point (tip).
• Standard position of a vector: Its initial point is at the origin.
• A vector's direction is the angle between the vector and the positive x-axis.
• Magnitude of a vector is proportional to the length of its representative line segment.
• A quadrant bearing is a directional measurement between 0° and 90° east or west of the north-south line.
• A true bearing is a directional measurement where the angle is measured clockwise from north, given using three digits.
• Parallel vectors have the same or opposite direction but not necessarily the same magnitude.
• Equivalent vectors have the same magnitude and direction.
• Opposite vectors have the same magnitude but opposite direction.
• Vector addition is done geometrically using the triangle method (tip-to-tail) or parallelogram method (tail-to-tail).

Vectors in the Coordinate Plane

• Vectors can be described using rectangular coordinates.
• Component Form: A vector OP is denoted by (x, y), with x and y being rectangular components.

Component Form of a Vector

• For vector AB with initial point A(x₁, y₁) and terminal point B(x₂, y₂), the component form is (x₂ - x₁, y₂ - y₁).

Magnitude of a Vector

• Magnitude: |v| = √(x₂ - x₁)² + (y₂ - y₁)² given the initial and terminal points,
• Magnitude: |v| = √(a² + b²) given vector components.

Vector Operations

• Vector Addition: a + b = (a₁ + b₁, a₂ + b₂), where a = (a₁, a₂) and b = (b₁, b₂).
• Vector Subtraction: a - b = (a₁ - b₁, a₂ - b₂), where a = (a₁, a₂) and b = (b₁, b₂).
• Scalar Multiplication: ka = (ka₁, ka₂), where a = (a₁, a₂) and k is a scalar.

Unit Vectors

• A vector with a magnitude of 1 is a unit vector.
• Any non-zero vector v can be expressed as a scalar multiple of a unit vector u.
• u = v / |v| (a unit vector in the same direction as v).

Vector Direction

• The direction of vector v = (a,b) is specified by the direction angle θ that v makes with the positive x-axis.
• Vector v can be written in component form as v = (|v| cos θ, |v| sin θ).
• Vector v can be written as a linear combination of unit vectors i and j: v = |v|(cos θ)i + |v|(sin θ)j.

Dot Product of Vectors

• The dot product is defined as a • b = a₁b₁ + a₂b₂ for vectors a = (a₁, a₂) and b = (b₁, b₂).
• The dot product yields a scalar, not a vector.

Orthogonal Vectors

• Vectors a and b are orthogonal if and only if a • b = 0.

Angle Between Two Vectors

• The angle θ between vectors a and b can be found using cos θ = (a • b) / (|a| |b|).

Projection of u onto v

• The projection of u onto v is denoted as projᵥu.
• projᵥu = ((u • v) / |v|²) * v.

Work Done by a Force

• Work done (W) by a force F is the dot product of the force and the displacement vector AB.
• W = F • AB = |F| |AB| cos θ.

Coordinates in Three Dimensions

• 3D space uses x, y, and z axes.
• A point in space is represented by an ordered triple: (x, y, z).

Distance and Midpoint Formulas in Space

• Distance between points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is: √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
• Midpoint formula M: ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)

Vectors in 3D

• For the directed line segment from A(x₁, y₁, z₁) to B(x₂, y₂, z₂), the component form is (x₂ - x₁, y₂ - y₁, z₂ - z₁).
• With vector components <a1, a2, a3> the magnitude |AB| : √(a₁² + a₂² + a₃²).

Dot Product and Orthogonal Vectors in Space

• Defined as a • b = a₁b₁ + a₂b₂ + a₃b₃, for vectors a = <a1, a2, a3> and b = <b1, b2, b3>.
• Vectors a and b are orthogonal if a • b = 0.

Cross Products

• The cross product of two vectors a and b (a x b), results in a vector.
• a x b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k
• The cross product (a x b) is perpendicular to both vectors a and b.

Torque

• Torque measures the effectiveness of a force causing rotation.
• Expressed as T = r x F, where r is a directed distance and F is the applied force measured in Newton meters(N × m).