Pre-Calculus — Essential Formulas

Pre-Calculus CBE one-pager — unit circle, trig identities, conic sections, sequences, and polynomial functions.

Unit Circle (Key Values)

• 0°: (1, 0) — sin = 0, cos = 1
• 30°: (√3/2, 1/2) — sin = 1/2, cos = √3/2
• 45°: (√2/2, √2/2)
• 60°: (1/2, √3/2)
• 90°: (0, 1) — sin = 1, cos = 0

Trig Identities

• Pythagorean: sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
• Double-angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ
• Half-angle: sin²(θ/2) = (1 − cos θ)/2
• tan θ = sin θ / cos θ

Inverse Trig

• arcsin: range [−π/2, π/2]
• arccos: range [0, π]
• arctan: range (−π/2, π/2)

Conic Sections

• Circle: (x − h)² + (y − k)² = r²
• Ellipse: (x − h)²/a² + (y − k)²/b² = 1
• Hyperbola: (x − h)²/a² − (y − k)²/b² = 1
• Parabola: y − k = a(x − h)² (vertical) or x − h = a(y − k)² (horizontal)

Polynomial Functions

• Degree n → at most n real zeros
• End behavior: even degree + positive lead → both ends up; odd degree + positive lead → down-left, up-right
• Rational zero theorem: rational zeros = ±(factor of constant) / (factor of leading coeff)

Exponential & Log Functions

• y = a·b^x: domain ℝ, range (0, ∞)
• y = log_b(x): domain (0, ∞), range ℝ
• Inverse pair: f(f⁻¹(x)) = x

Sequences & Series

• Arithmetic: a_n = a₁ + (n − 1)d
• Geometric: a_n = a₁·r^{n − 1}
• Sigma sum: Σ from i=1 to n of i = n(n + 1)/2
• Σ i² = n(n + 1)(2n + 1)/6

Limits (Intro)

• Direct substitution works when continuous
• If 0/0 form: factor, simplify, then substitute
• One-sided limits: lim from left ≠ lim from right ⇒ no two-sided limit

Common Test Mistakes

• Mixing up sine and cosine values on the unit circle
• Forgetting domain restrictions on inverse trig
• Confusing ellipse and hyperbola signs (+ vs −)
• Using arithmetic formula on a geometric sequence

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