Difficulty: Foundational | Prerequisites: High school algebra and geometry, basic right-triangle trigonometry.
Calculus 1 builds directly on top of algebra, coordinate geometry, and trigonometry. If those foundations are shaky, limits and derivatives will feel arbitrary rather than logical. This formula sheet collects the pre-calculus tools you are expected to know cold before the course even touches a limit. Treat these not as things to memorise from scratch, but as a checklist: if any formula here looks unfamiliar, go back and patch that gap before moving forward.
This covers the algebraic identities, geometric formulas, and trig identities you will use constantly in Calculus 1. You need the quadratic formula, the difference-of-squares and difference-of-cubes factorisations, basic area and volume formulas, the Pythagorean identity, double-angle formulas, and the standard trig-value table. These are the tools, not the calculus itself, but you cannot do the calculus without them.
Difference of squares
The identity a² − b² = (a − b)(a + b). In simple terms, any time you see one square minus another, you can factor it into two binomials. You will use this constantly when simplifying limits.
Difference of cubes
The identity a³ − b³ = (a − b)(a² + ab + b²). Think of it as the cubic cousin of difference of squares, with a trinomial factor instead of a binomial.
Quadratic formula
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. This is your fallback whenever factoring by inspection does not work.
Discriminant
The expression b² − 4ac under the radical in the quadratic formula. It tells you how many real roots a quadratic has: positive means two, zero means one (a repeated root), negative means none.
Pythagorean identity
sin²θ + cos²θ = 1. In simple terms, for any angle on the unit circle, the square of the sine plus the square of the cosine always equals one. This is the single most-used trig identity in calculus.
Double-angle formulas
sin(2θ) = 2 sin θ cos θ and cos(2θ) = cos²θ − sin²θ. These rewrite a function of 2θ in terms of θ. The cosine version has three equivalent forms, which is a common source of confusion.
Similar triangles
Triangles with the same angles, so their corresponding sides are proportional: AB/DE = BC/EF = AC/DF. In simple terms, one triangle is a scaled copy of the other, and you can set up ratios between matching sides.
Difference of squares: a² − b² = (a − b)(a + b)
Use case: simplifying rational expressions in limit problems where direct substitution gives 0/0.
Example: factor x² − 9 as (x − 3)(x + 3) to cancel a common factor.
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)
Note the middle term of the trinomial is positive (ab, not −ab). The sum of cubes has a³ + b³ = (a + b)(a² − ab + b²), with the signs flipped.
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
Works for any quadratic ax² + bx + c = 0.
The ± gives two solutions. When b² − 4ac = 0 they collapse to one repeated root.
Circle: Area = πr². Circumference = 2πr.
Equation of a circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r².
Recognising this form matters for related-rates problems and for sketching curves.
Distance formula: between (x₁, y₁) and (x₂, y₂) is √((x₁ − x₂)² + (y₁ − y₂)²).
This is just the Pythagorean theorem applied to a right triangle on the coordinate plane.
Area of a triangle: ½bh.
Trig ratios from a right triangle:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Similar triangles: If △ABC is similar to △DEF, then AB/DE = BC/EF = AC/DF. Corresponding sides are proportional.
Sphere: Volume = (4/3)πr³. Surface area = 4πr².
Cylinder or prism: Volume = (height)(area of base).
Cone or pyramid: Volume = (1/3)(height)(area of base).
Pythagorean identity: sin²θ + cos²θ = 1
Rearranges to sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ. Both rearrangements appear constantly.
Double-angle for sine: sin(2θ) = 2 sin θ cos θ
Double-angle for cosine (three equivalent forms):
cos(2θ) = cos²θ − sin²θ
cos(2θ) = 1 − 2 sin²θ
cos(2θ) = 2 cos²θ − 1
The second and third forms are derived by substituting the Pythagorean identity. They are used to derive the half-angle and power-reduction formulas you will meet later.
Standard trig values table:
x | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
sin(x) | 0 | 1/2 | √2/2 | √3/2 | 1 |
cos(x) | 1 | √3/2 | √2/2 | 1/2 | 0 |
tan(x) | 0 | √3/3 | 1 | √3 | DNE |
You should be able to reproduce this table from memory. Notice that the sine row reads 0, 1/2, √2/2, √3/2, 1, while the cosine row reads the same values in reverse. tan(π/2) does not exist because cos(π/2) = 0.
a^2 - b^2 = (a-b)(a+b)a^3 - b^3 = (a-b)(a^2 + ab + b^2)x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}(x-h)^2 + (y-k)^2 = r^2\sin^2\theta + \cos^2\theta = 1\sin(2\theta) = 2\sin\theta\cos\theta\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1The distance formula and circle equations are used whenever engineers or physicists model anything involving radius, range, or proximity, from GPS trilateration to antenna coverage. Volume and surface-area formulas for spheres, cylinders and cones underpin tank design, fluid storage calculations, and material-cost estimation in manufacturing. The trig identities show up in signal processing (Fourier analysis decomposes signals into sines and cosines) and in any physics problem involving oscillation or wave behaviour.
Students often confuse the difference of cubes with the difference of squares and try to factor a³ − b³ as (a − b)(a + b). It does not work. The cubic factorisation produces a trinomial factor.
The cosine double-angle formula has three forms. Students frequently memorise only one and then fail to recognise the others when they appear in a problem, or they mix up the signs.
Many students think sin(2θ) = 2 sin θ. It does not. The correct identity is sin(2θ) = 2 sin θ cos θ. Doubling the angle is not the same as doubling the function value.
Students sometimes apply the distance formula but forget to square the differences before adding, or they forget the square root at the end.
⚠️ Factoring (difference of squares, difference of cubes) is the primary technique for resolving 0/0 indeterminate forms in limit problems. If you cannot factor fluently, you will stall on limit questions.
⚠️ The trig values table is not provided on most MTH 132 exams. You are expected to know these values instantly.
⚠️ The Pythagorean identity and double-angle formulas appear in derivative and integral simplification throughout the course. They are not confined to the trig chapter.
⚠️ Similar-triangle setups appear in related-rates problems (shadow problems, sliding-ladder problems). You need the proportionality relationship without hesitation.
True or false: a³ − b³ = (a − b)(a² − ab + b²). _____
Fill in the blank: sin(2θ) = _____.
True or false: cos(π/3) = √3/2. _____
Fill in the blank: The volume of a cone is _____.
True or false: The distance between (1, 2) and (4, 6) is √(9 + 16) = 5. _____
Answers: 1. False (the middle term is +ab, not −ab). 2. 2 sin θ cos θ. 3. False (cos(π/3) = 1/2; you may be thinking of cos(π/6) = √3/2). 4. (1/3)(height)(area of base). 5. True.
Q: Factor x² − 16 completely.
A: (x − 4)(x + 4), by difference of squares.
Q: Factor 8x³ − 27.
A: (2x − 3)(4x² + 6x + 9), by difference of cubes with a = 2x and b = 3.
Q: Write all three forms of the cos(2θ) identity.
A: cos(2θ) = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1.
Q: Find the distance between the points (−1, 3) and (5, −1).
A: √((−1 − 5)² + (3 − (−1))²) = √(36 + 16) = √52 = 2√13.
Q: A right triangle has hypotenuse 10 and one leg of length 6. What is sin θ for the angle opposite the leg of length 6?
A: sin θ = opposite / hypotenuse = 6/10 = 3/5.
These algebraic factorisations connect directly to evaluating limits (the next major topic). When you encounter a 0/0 form in a limit, the fix is almost always to factor and cancel, and these identities are the tools for that. The trig identities connect to derivatives of trigonometric functions: proving that (sin x)' = cos x relies on the limit of sin θ / θ, and simplifying derivative results often requires the Pythagorean identity. The geometric formulas (area, volume, surface area) connect to optimisation and related-rates problems in later chapters, where you will set up equations using these formulas and then differentiate.
Difference of squares, difference of cubes, sum of cubes, quadratic formula, discriminant, completing the square, FOIL, factoring, circle equation, standard form of a circle, distance formula, Pythagorean theorem, area of a triangle, similar triangles, proportional sides, volume of a sphere, surface area, cylinder volume, cone volume, pyramid volume, sin cos tan, SOH CAH TOA, Pythagorean identity, double angle formula, trig values table, unit circle values, special angles, reference angles, MTH 132, Calculus 1, Michigan State University, pre-calculus review.