📐Precalculus & Calculus Trig

Trigonometric Identities & Unit Circle Formula Sheet

Every high-yield trigonometric identity for Precalculus, Calculus II trig integrals, and Physics vectors—organized with exam traps and 1-click self-test mode.

High-Yield Formulas

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Pythagorean Identity #1

sin²(θ) + cos²(θ) = 1

When to use: Simplifying trig expressions and converting between sin²(θ) and 1 - cos²(θ).

⚠️ Exam Trap: Remember this applies to squared terms only: sin(θ) + cos(θ) ≠ 1.

Secant-Tangent Pythagorean

1 + tan²(θ) = sec²(θ)

When to use: Trig substitution in Calculus II and simplifying sec/tan derivatives.

⚠️ Exam Trap: Divide sin²θ + cos²θ = 1 by cos²θ on scratch paper if you ever blank on the sign.

Sine Double-Angle Identity

sin(2θ) = 2 · sin(θ) · cos(θ)

When to use: Solving trig equations with 2θ and projectile range R = v₀²sin(2θ)/g.

⚠️ Exam Trap: Never factor out the 2 directly: sin(2θ) ≠ 2·sin(θ).

Cosine Double-Angle Identities

cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)

When to use: Power-reducing integrals and solving quadratic trig equations.

⚠️ Exam Trap: Watch the minus sign between cos²(θ) and sin²(θ).

Power-Reduction Formulas

sin²(θ) = (1 - cos(2θ))/2, cos²(θ) = (1 + cos(2θ))/2

When to use: Integrating ∫ sin²(x) dx or ∫ cos²(x) dx in Calculus.

⚠️ Exam Trap: sin²(θ) uses the minus sign (1 - cos 2θ), while cos²(θ) uses the plus sign.

Law of Cosines

c² = a² + b² - 2ab · cos(C)

When to use: Finding a missing side or angle in any non-right triangle (SAS or SSS).

⚠️ Exam Trap: When C = 90°, cos(90°) = 0 and it reduces to the Pythagorean Theorem.

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Frequently Asked Questions

How can I memorize the Unit Circle values in 2 minutes?

Use the √n / 2 pattern for 0°, 30°, 45°, 60°, 90°: sine goes √0/2, √1/2, √2/2, √3/2, √4/2, while cosine goes in reverse from √4/2 down to √0/2.

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