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11 multiple-choice questions and 15 flashcards on Applications of Trigonometry: Laws, Vectors, Polar, about 7% of the High School Pre-Calculus bank. Every one carries a written rationale.
Applications of Trigonometry: Laws, Vectors, Polar is one of 10 chapters in CoStudy's High School Pre-Calculus bank, and it holds 11 of the bank's 150 multiple-choice questions — roughly 7% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
8 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
Evaluate: cos(180°)
Answer: B — -1 (cos 180° = -1 from unit circle)
Unit circle: at 180°, point is (-1, 0). cos = x-coordinate = -1. sin = y-coordinate = 0. Memorize unit circle exact values. cos(0) = 1, cos(90°) = 0, cos(180°) = -1, cos(270°) = 0.
Find the angle between u = ⟨1, 0⟩ and v = ⟨1, 1⟩ (in degrees).
Answer: D — 45°
D) cos θ = (u · v)/(|u||v|) = 1/(1 · √2) = √2/2 → θ = 45°. A) Used a 30-60-90 triangle by mistake. B) Confused with the complement. C) Would require zero dot product.
The equation x² + y² − 6x + 8y = 0 represents a:
Answer: C — Circle
C) Complete the square: (x − 3)² + (y + 4)² = 25 → circle centered at (3, −4) with radius 5. A) Ellipse would have unequal coefficients on x² and y². B) Hyperbola has opposite signs. D) Parabola has only one squared term.
Identity: sin²(x) + cos²(x) = ?
Answer: E — 1 (Pythagorean identity)
Pythagorean identity: sin²x + cos²x = 1. Comes from unit circle: x² + y² = 1. Other identities derived: 1 + tan² = sec², 1 + cot² = csc². Used heavily in calculus integration.
Identity: sin(2x) = ?
Answer: D — 2sin(x)cos(x) (double-angle formula)
Double-angle formulas: sin(2x) = 2sin(x)cos(x). cos(2x) has three forms: cos²-sin², 2cos²-1, 1-2sin². Used to simplify trig expressions and evaluate at exact angles.
Convert polar r = 6 sin(θ) to rectangular form.
Answer: A — x² + y² = 6y, a circle
A) Multiply both sides by r: r² = 6 r sin θ → x² + y² = 6y. Completing the square: x² + (y − 3)² = 9 — circle centered (0, 3), radius 3. B) Treated sin(θ) as y/x. C) Confused with r = 6. D) Random.
Convert polar (r, θ) = (4, 60°) to rectangular:
Answer: E — (2, 2√3) (x = r·cos(θ) = 4·(1/2) = 2; y = r·sin(θ) = 4·(√3/2) = 2√3)
Polar-to-rectangular: x = r cos θ, y = r sin θ. r=4, θ=60°. cos 60° = 1/2, sin 60° = √3/2. x = 4·(1/2) = 2; y = 4·(√3/2) = 2√3. Result: (2, 2√3).
Convert (−2, 2) to polar (r > 0, 0 ≤ θ < 2π):
Answer: B — (2√2, 3π/4)
B) r = √(4 + 4) = 2√2. Point (−2, 2) lies in QII, so θ = π − π/4 = 3π/4. A) That is QI. C) That is QIII. D) That is QIV. arctan alone is not enough — quadrant matters.
4 cards from the 15 in this chapter.
How do you find a unit vector in the direction of v?
Divide v by its magnitude: v / |v|.
What is a unit vector?
A vector with magnitude 1, pointing in a particular direction.
What is a matrix?
A rectangular array of numbers arranged in rows and columns. Used for linear systems, transformations, and more.
What is the angle between vectors u and v in terms of dot product?
cos θ = (u·v) / (|u||v|).
These are a sample. The full Applications of Trigonometry: Laws, Vectors, Polar chapter runs 26 items with per-chapter progress tracking, on the web and in the iOS app.
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