Master trigonometry with this comprehensive flashcard deck covering every key concept from basic SOH CAH TOA ratios to the unit circle, trigonometric identities, and the graphs of sine, cosine, and tangent. Trigonometry is essential for physics, engineering, a...
Master trigonometry with this comprehensive flashcard deck covering every key concept from basic SOH CAH TOA ratios to the unit circle, trigonometric identities, and the graphs of sine, cosine, and tangent. Trigonometry is essential for physics, engineering, architecture, and
navigation — and it is a major component of the SAT, ACT, A-Level, AP Precalculus, and AP Calculus exams.
This deck covers trigonometric ratios, exact values, the unit circle, solving triangles with the sine and cosine rules, amplitude and period of trig graphs, the Pythagorean identities, double angle formulas, inverse trigonometric functions, and general solutions. Based on publicly available resources including OpenStax Precalculus and CK-12.
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A memory aid for trig ratios in a right triangle: Sin = Opposite/Hypotenuse. Cos = Adjacent/Hypotenuse. Tan = Opposite/Adjacent.
sin 0° = 0, cos 0° = 1, tan 0° = 0.
sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3.
sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1.
sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
sin 90° = 1, cos 90° = 0, tan 90° = undefined (division by zero).
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.
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.
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.
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.
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.
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).
sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. tan 2θ = 2tan θ / (1 − tan²θ).
Period of tan(x) = π (not 2π). Tan has vertical asymptotes at x = π/2 + nπ for all integers n. It has no amplitude.
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.
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).
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.
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.
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.
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.
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).
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.
