1. Fundamental Trigonometric Ratios & Reciprocal Relations

In a right-angled triangle with angle θ, opposite side (Perpendicular p), adjacent side (Base b), and Hypotenuse h: sin θ = p/h, cos θ = b/h, tan θ = p/b. Reciprocal ratios: csc θ = 1/sin θ = h/p, sec θ = 1/cos θ = h/b, cot θ = 1/tan θ = b/p. Quotient identities: tan θ = sin θ / cos θ, and cot θ = cos θ / sin θ.

2. Standard Angle Values Table (0°, 30°, 45°, 60°, 90°)

Core values: sin(0°) = 0, sin(30°) = 1/2, sin(45°) = 1/√2, sin(60°) = √3/2, sin(90°) = 1. Cosine values reverse this sequence: cos(0°) = 1, cos(30°) = √3/2, cos(45°) = 1/√2, cos(60°) = 1/2, cos(90°) = 0. Tangent values: tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3, tan(90°) is undefined.

3. The ASTC Quadrant Rule ("All-Silver-Tea-Cups")

In coordinate planes: Quadrant I (0°–90°): ALL ratios are positive. Quadrant II (90°–180°): Only SINE and cosecant are positive. Quadrant III (180°–270°): Only TANGENT and cotangent are positive. Quadrant IV (270°–360°): Only COSINE and secant are positive. Use (180° ± θ) or (360° ± θ) to maintain ratio names, whereas (90° ± θ) and (270° ± θ) switch ratios (sin ↔ cos, tan ↔ cot, sec ↔ csc).

4. Core Pythagorean Identities

The three fundamental identities derived from the Pythagorean theorem: (1) sin² θ + cos² θ = 1, leading to sin² θ = 1 - cos² θ and cos² θ = 1 - sin² θ; (2) 1 + tan² θ = sec² θ, meaning sec² θ - tan² θ = 1; (3) 1 + cot² θ = csc² θ, meaning csc² θ - cot² θ = 1. These identities appear in over 70% of board and competitive exam simplification problems.

5. Compound Angle & Double Angle Identities

Addition and subtraction: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B (note the sign reversal); tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B). Double angle formulas: sin 2θ = 2 sin θ cos θ = (2 tan θ) / (1 + tan² θ); cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ = (1 - tan² θ) / (1 + tan² θ).

6. Teacher's Exam Hall Hack: The 45° Substitution Test

When faced with a complex trigonometric equation in a multiple-choice exam, substitute θ = 45° (or 30° if 45° creates an undefined denominator). Evaluating the arithmetic values of the options against the question takes under 20 seconds and completely eliminates tedious algebraic derivations.