Trigonometry
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.
All Cards in This Deck
Question
What is SOH CAH TOA?
Answer
A memory aid for trig ratios in a right triangle: Sin = Opposite/Hypotenuse. Cos = Adjacent/Hypotenuse. Tan = Opposite/Adjacent.
Question
What is the exact value of sin, cos, and tan at 0°?
Answer
sin 0° = 0, cos 0° = 1, tan 0° = 0.
Question
What is the exact value of sin, cos, and tan at 30°?
Answer
sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3.
Question
What is the exact value of sin, cos, and tan at 45°?
Answer
sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1.
Question
What is the exact value of sin, cos, and tan at 60°?
Answer
sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
Question
What is the exact value of sin, cos, and tan at 90°?
Answer
sin 90° = 1, cos 90° = 0, tan 90° = undefined (division by zero).
Question
What is the unit circle?
Answer
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.
Question
What is the Pythagorean identity?
Answer
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.
Question
What are radians and how do they convert to degrees?
Answer
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.
Question
What are the common degree-radian equivalents?
Answer
0°=0, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 120°=2π/3, 180°=π, 270°=3π/2, 360°=2π.
Question
What is the sine rule?
Answer
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.
Question
What is the cosine rule?
Answer
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.
Question
What is the ASTC / CAST rule?
Answer
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).
Question
What are the double angle formulas?
Answer
sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. tan 2θ = 2tan θ / (1 − tan²θ).
Question
What is the period of tan(x)?
Answer
Period of tan(x) = π (not 2π). Tan has vertical asymptotes at x = π/2 + nπ for all integers n. It has no amplitude.
Question
What are reciprocal trig functions?
Answer
Cosecant: csc θ = 1/sin θ. Secant: sec θ = 1/cos θ. Cotangent: cot θ = 1/tan θ = cos θ/sin θ.
Question
What are inverse trig functions and their ranges?
Answer
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.
Question
What are the trig addition formulas?
Answer
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).
Question
What is the graph of sin(x) key features?
Answer
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.
Question
What is the graph of cos(x) key features?
Answer
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.
Question
What are co-function identities?
Answer
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.
Question
What is the area of a triangle using trig?
Answer
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.
Question
How do you solve trig equations like sin x = ½?
Answer
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).
Question
What is Pythagoras in 3D and how is it used?
Answer
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.