Commonly Used Formulas

Here is a collection of commonly used mathematical formulas, grouped by topic, and linked to the location in the text where they are introduced.  Each includes a description in English.  Remember: To apply a formula to a given expression, you should:

1. match up the variables on one side of the formula you wish to use with the corresponding parts of the given expression to obtain values for each of the variables in the formula,
2. substitute on both sides of the formula with the values from part a),
3. substitute the other side of the formula into the given expression.

Note: Any formula may be used "in either direction"; that is, you can match the left-side and substitute the right, or vice-versa.

Geometry

• If a line y = a x + b, goes through the two points (x₁, y₁) and (x₂, y₂), then , and b = the y-value where the line crosses the y-axis (i.e., when x = 0): this is often referred to as the "slope-intercept form" of  a line.
• The distance between two points a and b on the number line is given by |a - b|, where |x| is the absolute value function: the distance formula for points on a line.
• If a right triangle has sides of length a, b, and c, as in the following diagram:

  

  then a² + b² = c²: this is the Pythagorean theorem.

• The distance d between two points (x₁, y₁) and (x₂, y₂) in the plane is given by the equation : this is the distance formula in the plane.

Note: Using the distance formula on the horizontal and vertical number lines, respectively:

(and the fact that |z|² = z²), this equations follows from applying the square root to both sides of the equation of the Pythagorean Theorem, with c = d, a = |y₁ - y₂|, and b = |x₁ - x₂| . 

• The equation for an arbitrary point (x, y) circle of radius r centered at the point (a, b) is given by the equation (x - a)² + (y - b)² = r²: since the distance from the center to a point on the circle is equal to the radius, this equation comes from squaring both sides of distance formula in the plane.
• The area, A, of a circle of radius r is A = pr².  That is because:
  • all circles are the result of scaling the circle of radius 1 by a factor of r,
  • the area of a circle of radius 1 is p, and
  • scaling by r changes the area of any plane figure by a factor of r².
• The circumference of (i.e., length around) a circle of radius r is 2pr.  That is because:
  • all circles are the result of scaling the circle of radius 1 by a factor of r,
  • the area, A, of a symmetric plane figure (such as a square, hexagon, circle, etc.) is equal to its circumference, C, times half the distance, r, from the center to a "side".  Note: This follows from the area formula for triangles A = 1/2hb.  In other words, A = 1/2Cr, so that for circles, pr² = 1/2Cr, and C = 2pr.

  Note: As you might expect, this formula reflects the fact that scaling by r changes the length of any plane figure by a factor of r.

Algebra

• x(y + z) = xy + xz, (x + y)(z + w) = xz + xw + yz + yw, etc.: "When multiplying sums, you must add every possible product"; this is called the distributive law.
• If ax² + bx + c = 0, then : "The quadratic formula".
• : "The absolute value function gives the magnitude of a number, i.e., as a positive value."

Note: We may also describe the absolute value function algebraically as .

• |x y| = |x| |y|: "The absolute value 'preserves' products ."

Exponents

• x² = xx, x³ = xxx, etc.: "Whole number powers are repeated products." 
• x⁵x³ = x⁸, x⁸/x³ = x⁵ "When multiplying/dividing, exponents add/subtract." 
• (3y⁵z)² = 3²(y⁵)²z², etc.: "Powers distribute across multiplication." 
• (y⁵)² = y¹⁰, etc.: "Successive powers multiply." 

Note: While the first "rule" is actually a definition, the last three rules follow logically from the first and basic rules of arithmetic.

• x⁰ = 1: "Any exponential takes 0 to 1"  
• x^-2 = 1/x², (x/y)^-3/2 = (y/x)^3/2: "A negative exponent stands for reciprocation."  
• : "A reciprocal in an exponent is like a root."  

Note: These rules are also definitions, designed to ensure that the previous rules still hold even when the exponents are no longer positive whole numbers.  

• : " An 'ordinary' radical is always assumed to be a square root." 

Logarithms

• log_b = exp_b^-1, where exp_b is the exponential function with base b, exp_b(x) = b^x: "Logarithms are the inverses of exponentials." 

Note: This rule is actually a definition of what we mean by a "logarithmic function with base b".

• log_b(b^x) = x and : "Logarithms and exponentials cancel each other out." 

Note: Because of the characteristic property of inverses, the previous rule is exactly the same as the one before it; it simply looks different algebraically and in English. 

• log = log₁₀: "If you don't see a base, you can assume it is base 10." 
• ln = log_e: "The 'natural' logarithm uses base e." 

Note: These rules are also definitions.  

Notice that the following rules are exactly the reverse of the corresponding rules of exponents.  

• log_b(1) = 0: "Any logarithm takes 1 to 0"  
• log_b(s) + log_b(t) =  log_b(st), log_b(s) - log_b(t) =  log_b(s/t): "When adding/subtracting logarithms, multiply/divide the inputs." 
• log_b(st) = log_b(s) + log_b(t), log_b(s/t) = log_b(s) - log_b(t): "Logarithms turn products/quotients into sums/differences." 

Note: The previous rule is exactly the same as the one before it; it just sounds different when you say it in English. 

• log_b(s^t) =  t log_b(s): "Powers can be pulled outside a logarithm as a multiple." 

Note: This rule corresponds to the rule of repeated exponents.

• log_c(x) =  log_c(b) log_b(x) and : "The change-of-base formulas."
•  log_1/b(x) = log_b(1/x) = -log_b(x) and : "Reciprocation of the input corresponds to negation of the output or reciprocation of the base."

Complex Numbers

• : "Complex conjugation replaces i by -i." 
• , , , and for complex numbers x = a + b i and y = c + d i (where a, b, c, and d are all real numbers) "Complex conjugation preserves all arithmetic operations on complex numbers." 
• , where x and y are both real numbers: "The distance formula using complex arithmetic." 

Trigonometry

• e^{t i} = cos(t) + sin(t) i: "Euler's formula." 
• 1 = cos²(t) + sin²(t) : "The Pythagorean identity." 
• csc(t) = 1/sin(t), sec(t) = 1/cos(t), cot(t) = cos(t)/sin(t), tan(t) = sin(t)/cos(t): "Basic definitions."  
• cos(t) = sin(p/2 - t), csc(t) = sec(p/2 - t), cot(t) = tan(p/2 - t): "Co-functions are functions of complementary angles."
• cos(t ± s) = cos(t)cos(s) sin(t)sin(s), sin(t ± s) = sin(t)cos(s) ± cos(t)sin(s), tan(t ± s) = (tan(t) ± tan(s))/(1 tan(t)tan(s)): "The angle sum/difference formulas." 
• cos(2t) = cos²(t) - sin²(t) = 2cos²(t) - 1 = 1 - 2sin²(t), sin(2t) = 2sin(t)cos(t), tan(2t) = 2tan(t)(1-tan²(t)): "The double-angle formulas." 
• , , : "The half-angle formulas." 

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Glossary