What is SOH CAH TOA? — A memory aid for trig ratios in a right triangle: Sin = Opposite/Hypotenuse. Cos = Adjacent/Hypotenuse. Tan = Opposite/Adjacent.
What is the exact value of sin, cos, and tan at 0°? — sin 0° = 0, cos 0° = 1, tan 0° = 0.
What is the exact value of sin, cos, and tan at 30°? — sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3.
What is the exact value of sin, cos, and tan at 45°? — sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1.
What is the exact value of sin, cos, and tan at 60°? — sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
What is the exact value of sin, cos, and tan at 90°? — sin 90° = 1, cos 90° = 0, tan 90° = undefined (division by zero).
What is the unit circle? — A circle with radius 1 centred at the origin. Any point on it is (cos θ, sin θ) where θ is the angle from the positive x-axis. It defines trig functions for all angles.
What is the Pythagorean identity? — sin²θ + cos²θ = 1 — true for all angles. Derived from the Pythagorean theorem on the unit circle. From this: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.
What are radians and how do they convert to degrees? — Radians measure angles by the arc length on a unit circle. π radians = 180°. Convert degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
What are the common degree-radian equivalents? — 0°=0, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 120°=2π/3, 180°=π, 270°=3π/2, 360°=2π.
What is the sine rule? — a/sin A = b/sin B = c/sin C where a,b,c are side lengths and A,B,C are the opposite angles. Use when you know: two angles + one side, or two sides + one non-included angle.
What is the cosine rule? — a² = b² + c² − 2bc cos A. Use when you know: two sides + included angle, or all three sides (to find an angle). Rearranged for angle: cos A = (b²+c²−a²) / 2bc.
What is the ASTC / CAST rule? — Tells which trig functions are positive in each quadrant: Q1: All positive. Q2: Sin positive. Q3: Tan positive. Q4: Cos positive. Memory: All Students Take Calculus (Q1→Q2→Q3→Q4).
What are the double angle formulas? — sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. tan 2θ = 2tan θ / (1 − tan²θ).
What is the period of tan(x)? — Period of tan(x) = π (not 2π). Tan has vertical asymptotes at x = π/2 + nπ for all integers n. It has no amplitude.
What are reciprocal trig functions? — Cosecant: csc θ = 1/sin θ. Secant: sec θ = 1/cos θ. Cotangent: cot θ = 1/tan θ = cos θ/sin θ.
What are inverse trig functions and their ranges? — arcsin (sin⁻¹): range [−π/2, π/2]. arccos (cos⁻¹): range [0, π]. arctan (tan⁻¹): range (−π/2, π/2). They give the angle whose trig ratio equals the input.
What are the trig addition formulas? — sin(A±B) = sin A cos B ± cos A sin B. cos(A±B) = cos A cos B ∓ sin A sin B. tan(A±B) = (tan A ± tan B) / (1 ∓ tan A tan B).
What is the graph of sin(x) key features? — Starts at (0,0). Reaches maximum of 1 at π/2. Returns to 0 at π. Reaches minimum of −1 at 3π/2. Returns to 0 at 2π. Period = 2π. Smooth, continuous, wave-like.
What is the graph of cos(x) key features? — Starts at (0,1). Crosses zero at π/2. Reaches minimum −1 at π. Returns to 0 at 3π/2. Returns to 1 at 2π. Period = 2π. Same shape as sin(x) shifted left by π/2.
What are co-function identities? — sin(90°−θ) = cos θ and cos(90°−θ) = sin θ. Also: tan(90°−θ) = cot θ. These show the relationship between complementary angle functions.
What is the area of a triangle using trig? — A = ½ab sin C where a and b are two sides and C is the included angle between them. This extends the ½bh formula when the height is not directly known.
How do you solve trig equations like sin x = ½? — Find the principal value: x = 30° (or π/6). Then use CAST/ASTC for all solutions in the given domain. General solution: x = 30° + 360°n and x = 150° + 360°n (where sin is also positive in Q2).
What is Pythagoras in 3D and how is it used? — For a 3D rectangular box with dimensions l, w, h, the space diagonal = √(l²+w²+h²). Apply Pythagoras twice — first across the base, then up to the top corner.
