Accuplacer Advanced Algebra Formulas

Point Slope Form

• Represents the equation of a line using a known point (x1, y1) and slope m: ( y - y_1 = m(x - x_1) )

Trigonometric Ratios

• Sine (sin): Ratio of the length of the opposite side to the hypotenuse in a right triangle: ( \text{sin} = \frac{\text{opposite}}{\text{hypotenuse}} )
• Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse: ( \text{cos} = \frac{\text{adjacent}}{\text{hypotenuse}} )
• Tangent (tan): Ratio of sine to cosine: ( \text{tan} = \frac{\text{opposite}}{\text{adjacent}} )

Difference of Cubes Formula

• Can be factored as: ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) )

Distance and Midpoint Formulas

• Point Distance Formula: Calculates distance between two points ((x_1, y_1)) and ((x_2, y_2)): ( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} )
• Midpoint Formula: Finds the midpoint of a segment connecting two points: ( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) )

Area Formulas

• Area of Square: ( A = s^2 )
• Area of Rectangle: ( A = lw )
• Area of Parallelogram: ( A = bh )
• Area of Triangle: ( A = \frac{1}{2}bh ) or ( A = \frac{bh}{2} )
• Area of Circle: ( A = \pi r^2 )
• Area of Trapezoid: ( A = \frac{1}{2}(b_1 + b_2)h )
• Area of Equilateral Triangle: ( \frac{s^2 \sqrt{3}}{4} )
• Area of Hexagon: ( \frac{3\sqrt{3}}{2}a^2 )

Circumference and Volume Formulas

• Circumference of Circle: ( C = 2\pi r )
• Volume of Cube: ( V = s^3 )
• Volume of Rectangular Prism: ( V = lwh )
• Volume of Cone: ( V = \frac{1}{3} \pi r^2 h )
• Volume of Sphere: ( V = \frac{4}{3} \pi r^3 )
• Volume of Cylinder: ( V = \pi r^2 h )
• Volume of Pyramid: ( V = \frac{1}{3} lwh )

Exponential Functions

• Exponential Growth Formula: ( y = a(1 + r)^t )
• Exponential Decay Formula: ( y = a(1 - r)^t )

Trigonometric Identities

• Pythagorean Identities:
  • ( \sin^2 x + \cos^2 x = 1 )
  • ( \tan^2 x + 1 = \sec^2 x )
  • ( 1 + \cot^2 x = \csc^2 x )
• Double Angle Formulas:
  • ( \sin(2x) = 2 \sin x \cos x )
  • ( \cos(2x) = \cos^2 x - \sin^2 x )
  • ( \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x} )

Geometric Angle Formulas

• Inscribed Angle: Measures half of the arc it intercepts.
• Angle Formed by a Tangent and a Chord: Equal to half the measure of the intercepted arc.
• Angle Formed by Intersection of Two Chords: Average of measures of the arcs intercepted.
• Angle Outside the Circle: Difference between larger and smaller arcs, divided by 2.

Slope Relationships

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are opposite reciprocals, e.g., if one slope is ( 5 ), the other is ( -\frac{1}{5} ).

Job and Mixture Formulas

• Work Rate Formula: ( \text{Job} = \text{Work Rate} \times \text{Time} )
• Mixture Formula: ( (\text{percentage of solution}) \times (\text{volume of solution}) = \text{amount of liquid} )

Unit Circle Values

• ( x ) and ( y ) coordinates are represented by ( ( \cos, \sin ) ) respective to angles.