{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Laws of Sines & Cosines",
    "author": "Staples Education",
    "description": "Choosing between the two laws, the trigonometric area formula, and the ambiguous case. Authored: the whiteboard archive covers SOHCAHTOA and the special triangles thoroughly but never reaches the oblique-triangle laws.",
    "created": "2026-08-06T04:14:56.055Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "trig-laws",
      "title": "Laws of Sines & Cosines",
      "blurb": "Choosing between the two laws, solving ASA/AAS/SSS/SAS triangles, the trigonometric area formula, and the ambiguous case of SSA.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "laws-of-sines-cosines"
      ],
      "questions": [
        {
          "id": "tls-001",
          "topic": "Law of Sines",
          "difficulty": "medium",
          "stem": "In △ABC, m∠A = 30°, side a = 6, and m∠B = 45°. Find side b.",
          "choices": [
            "6√2",
            "3√2",
            "12",
            "6√3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Law of Sines says a/sin A = b/sin B. So 6/sin 30° = b/sin 45°, giving 6/0.5 = 12 as the common ratio, and b = 12 · sin 45° = 12(√2/2) = 6√2 ≈ 8.49.",
            "3√2 halves the correct answer — it uses 6 as the common ratio rather than 12. Dividing 6 by sin 30° = 0.5 doubles it.",
            "12 is the common ratio a/sin A itself, not a side. It still has to be multiplied by sin B.",
            "6√3 would follow from using a 60° angle. sin 45° = √2/2, which carries the √2."
          ],
          "hint": "Compute a/sin A first — that single number is the same for all three sides. Then multiply it by sin B."
        },
        {
          "id": "tls-002",
          "topic": "Law of Cosines",
          "difficulty": "medium",
          "stem": "A triangle has sides of 5 and 8 with an included angle of 60°. Find the third side.",
          "choices": [
            "7",
            "√89",
            "13",
            "√129"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Law of Cosines gives c² = a² + b² − 2ab·cos C = 25 + 64 − 2(5)(8)cos 60° = 89 − 80(0.5) = 89 − 40 = 49, so c = 7.",
            "√89 drops the −2ab·cos C term entirely, which is the same as assuming C = 90°. At 60° the correction is a real −40.",
            "13 adds the two sides. The third side of a triangle is always strictly less than that.",
            "√129 ADDS 40 instead of subtracting it, which is what happens if the sign of the cosine term is flipped. An angle under 90° makes the third side shorter than the right-angle case, not longer."
          ],
          "hint": "Law of Cosines is Pythagoras plus a correction term. Work out whether a 60° angle makes the opposite side longer or shorter than the right-angle case.",
          "diagram": "<svg viewBox=\"0 0 240 142\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"30,110 200,110 96,32\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><text x=\"58\" y=\"74\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5</text><text x=\"158\" y=\"66\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">8</text><text x=\"108\" y=\"130\" fill=\"#c0392b\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">c = ?</text><text x=\"96\" y=\"56\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">60°</text></svg>",
          "diagramCaption": "The 60° angle sits between the two known sides. Not drawn to scale"
        },
        {
          "id": "tls-003",
          "topic": "Choosing a Law",
          "difficulty": "medium",
          "stem": "You know all three side lengths of a triangle and want an angle. Which law applies?",
          "choices": [
            "Law of Cosines",
            "Law of Sines",
            "Either works equally well",
            "Neither — three sides do not determine the angles"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Law of Cosines relates three sides and one angle, so with SSS you can rearrange it to cos C = (a² + b² − c²)/(2ab) and solve for any angle directly.",
            "The Law of Sines pairs each side with the angle OPPOSITE it, so it needs at least one angle already known. With SSS you have none, and the equation has two unknowns.",
            "They are not interchangeable. Which one applies is decided by what you have: SSS and SAS need the Law of Cosines; ASA, AAS and SSA need the Law of Sines.",
            "Three sides absolutely determine the angles — that is exactly what SSS congruence means."
          ],
          "hint": "One law connects three sides and one angle; the other connects two sides and two opposite angles. Which fits SSS?"
        },
        {
          "id": "tls-004",
          "topic": "The Ambiguous Case",
          "difficulty": "hard",
          "stem": "Given a = 8, b = 10 and m∠A = 40°, how many triangles are possible?",
          "choices": [
            "Exactly one",
            "Two",
            "None",
            "Infinitely many"
          ],
          "correctIndex": 1,
          "explanations": [
            "Exactly one would need the configuration to be unambiguous. This is SSA — two sides and a non-included angle — and the side opposite the known angle is the SHORTER one, which is the setup that can close two ways.",
            "Correct. Two. Check the height from C to line AB: h = b·sin A = 10·sin 40° ≈ 6.43. Since h < a < b (6.43 < 8 < 10), side a is long enough to reach the base but short enough to reach it twice — once with an acute angle B and once with its obtuse supplement. This is the ambiguous case of the Law of Sines, the trigonometric face of why SSA is not a congruence criterion.",
            "None would require a < h, meaning side a is too short to reach the base at all. Here 8 > 6.43, so it reaches.",
            "Infinitely many would need an angle-only condition, like AAA. Fixing two lengths pins the triangle down to finitely many options."
          ],
          "hint": "Compute h = b·sin A, the shortest distance from C to the base line, then compare it with a and with b. Three comparisons, three outcomes."
        },
        {
          "id": "tls-005",
          "topic": "Area with Trigonometry",
          "difficulty": "medium",
          "stem": "A triangle has sides of 7 and 10 with an included angle of 30°. What is its area?",
          "choices": [
            "17.5",
            "35",
            "70",
            "8.75"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Area = ½ab·sin C = ½(7)(10)sin 30° = 35(0.5) = 17.5.",
            "35 is ½ab without the sine factor, which would be the area only if C were 90°.",
            "70 is the full product ab. That is twice the area of the RECTANGLE on those sides, let alone the triangle.",
            "8.75 halves the answer once too often — perhaps by applying both the ½ and a further ½ from sin 30° twice over."
          ],
          "hint": "Same formula as the parallelogram area, halved: a triangle is half the parallelogram on the same two sides."
        },
        {
          "id": "tls-006",
          "topic": "Law of Sines",
          "difficulty": "medium",
          "stem": "In △ABC, m∠A = 50°, m∠B = 60° and side a = 9. Which setup finds side c?",
          "choices": [
            "c = 9·sin 70° / sin 50°",
            "c = 9·sin 50° / sin 70°",
            "c = 9·sin 60° / sin 50°",
            "c² = 9² + b² − 2(9)(b)cos 70°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. First find ∠C = 180° − 50° − 60° = 70°. Then the Law of Sines gives c/sin C = a/sin A, so c = a·sin C/sin A = 9·sin 70°/sin 50°.",
            "This inverts the ratio, dividing by the sine of the angle opposite the side you WANT rather than the one you know. It would make c smaller than a, but ∠C > ∠A so c must be larger.",
            "sin 60° pairs with side b, not side c. Each side goes with the angle directly opposite it.",
            "The Law of Cosines is not wrong here, but it introduces b, which is unknown. Using it would mean solving for b first — extra work when the Law of Sines answers directly."
          ],
          "hint": "Find the third angle first. Then pair each side with the angle across from it."
        },
        {
          "id": "tls-007",
          "topic": "Law of Cosines",
          "difficulty": "hard",
          "stem": "A triangle has sides 6, 7 and 9. Find the cosine of the angle opposite the side of length 9.",
          "choices": [
            "1/21",
            "2/21",
            "11/21",
            "−1/21"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Rearranged, cos C = (a² + b² − c²)/(2ab) with c = 9 opposite C: (36 + 49 − 81)/(2·6·7) = 4/84 = 1/21. Positive but tiny, so C is just under 90° — consistent with 6² + 7² = 85 being only slightly more than 9² = 81.",
            "2/21 would come from 4/42, halving the denominator. 2ab = 2(6)(7) = 84, not 42.",
            "11/21 comes from (36 + 49 + 81 − ...) or a sign slip in the numerator. The side opposite the angle is SUBTRACTED.",
            "−1/21 flips the sign, which would make C obtuse. Since 6² + 7² > 9², the angle is acute and the cosine is positive."
          ],
          "hint": "Rearrange c² = a² + b² − 2ab·cos C for cos C, and be careful which side is opposite the angle you want."
        },
        {
          "id": "tls-008",
          "topic": "Choosing a Law",
          "difficulty": "easy",
          "stem": "You know two angles and the side between them (ASA). Which law finds another side most directly?",
          "choices": [
            "Law of Sines",
            "Law of Cosines",
            "The Pythagorean theorem",
            "Neither applies to ASA"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Two angles give the third for free, so every angle is known along with one side — exactly the pairing the Law of Sines needs.",
            "The Law of Cosines needs at least two sides. Here only one is known, so it cannot be applied directly.",
            "The Pythagorean theorem applies only to right triangles, and nothing in ASA guarantees one.",
            "ASA is a standard solvable case. It is one of the configurations the Law of Sines exists for."
          ],
          "hint": "Count what you have: how many angles, how many sides? Then match that to what each law consumes."
        },
        {
          "id": "tls-009",
          "topic": "The Ambiguous Case",
          "difficulty": "hard",
          "stem": "Given a = 5, b = 12 and m∠A = 40°, how many triangles are possible?",
          "choices": [
            "Two",
            "Exactly one",
            "None",
            "Three"
          ],
          "correctIndex": 2,
          "explanations": [
            "Two requires h < a < b. Here a = 5 is smaller than h ≈ 7.71, so it never reaches the base even once.",
            "Exactly one would need a ≥ b, letting the side swing past the foot of the perpendicular. Here a is much smaller than b.",
            "Correct. None. The height from C to line AB is h = b·sin A = 12·sin 40° ≈ 7.71. Side a = 5 is shorter than that, so it cannot reach the base line at all and no triangle closes.",
            "Three is never an outcome of the ambiguous case. SSA yields zero, one, or two triangles."
          ],
          "hint": "Compute h = b·sin A and compare it to a. If the side is shorter than the perpendicular distance, it cannot reach."
        },
        {
          "id": "tls-010",
          "topic": "Applications",
          "difficulty": "medium",
          "stem": "Two roads leave a junction at an angle of 70°. One car drives 4 km along the first and another drives 6 km along the second. How far apart are they? (cos 70° ≈ 0.342)",
          "choices": [
            "√(52 − 48(0.342))",
            "√(52 + 48(0.342))",
            "10 km",
            "2 km"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The two roads and the line between the cars form a triangle with the 70° angle included between the 4 km and 6 km sides. The Law of Cosines gives d² = 4² + 6² − 2(4)(6)cos 70° = 16 + 36 − 48(0.342) = 52 − 16.4 ≈ 35.6, so d ≈ 5.97 km.",
            "Adding the cosine term instead of subtracting it would make the cars further apart than a right-angle turn would put them, which is backwards — an angle under 90° brings them closer.",
            "10 km is 4 + 6, the distance if the cars drove in exactly opposite directions (a 180° angle). At 70° they are much closer.",
            "2 km is 6 − 4, the distance if they drove along the SAME road in the same direction. That is the 0° case."
          ],
          "hint": "The junction angle sits between the two distances travelled. That is the included-angle setup."
        },
        {
          "id": "tls-011",
          "topic": "Law of Sines",
          "difficulty": "medium",
          "stem": "In a triangle, the largest angle is always opposite which side?",
          "choices": [
            "The longest side",
            "The shortest side",
            "The side of average length",
            "It depends on the triangle"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Law of Sines makes each side proportional to the sine of its opposite angle, so bigger angles sit across from longer sides. It is also visible directly: opening an angle wider pushes the opposite side further apart.",
            "The shortest side sits opposite the SMALLEST angle — the same rule, read the other way.",
            "The average-length side sits opposite the middle angle, not the largest.",
            "It does not depend on the triangle. The side-angle ordering holds in every triangle without exception."
          ],
          "hint": "Picture opening one angle of a triangle wider while keeping the two sides fixed. What happens to the side across from it?"
        },
        {
          "id": "tls-012",
          "topic": "Law of Cosines",
          "difficulty": "easy",
          "stem": "For which angle C does the Law of Cosines c² = a² + b² − 2ab·cos C reduce to the Pythagorean theorem?",
          "choices": [
            "90°",
            "0°",
            "60°",
            "180°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. cos 90° = 0, so the −2ab·cos C term vanishes and the formula becomes c² = a² + b². The Pythagorean theorem is the right-angle special case of the Law of Cosines.",
            "At 0° the cosine is 1, giving c² = a² + b² − 2ab = (a − b)², so c = |a − b| — the degenerate case where the two sides lie along each other.",
            "At 60° the cosine is ½, leaving a correction of −ab. That is a real term, not zero.",
            "At 180° the cosine is −1, giving c² = (a + b)² — the other degenerate case, where the sides point in opposite directions."
          ],
          "hint": "The extra term disappears when the cosine is zero. At what angle is that?"
        },
        {
          "id": "tls-013",
          "topic": "Choosing a Law",
          "difficulty": "medium",
          "stem": "You are given two sides of a triangle and the angle between them (SAS), and asked for the third side. Which law applies?",
          "choices": [
            "Law of Sines",
            "Law of Cosines",
            "Either works equally well",
            "Neither — use the Pythagorean Theorem"
          ],
          "correctIndex": 1,
          "explanations": [
            "The Law of Sines needs an angle PAIRED WITH ITS OPPOSITE SIDE to form a usable ratio. In SAS the known angle sits between the known sides, so its opposite side is exactly the unknown — there is no complete ratio to start from.",
            "Correct. The Law of Cosines, c² = a² + b² − 2ab·cos C, is built for SAS: the included angle and its two adjacent sides are precisely its inputs, and it returns the opposite side directly.",
            "They are not interchangeable. Each law requires a specific pattern of given information, and SAS supplies what the Law of Cosines needs but not what the Law of Sines needs.",
            "The Pythagorean Theorem applies only to RIGHT triangles. Using it on a general triangle is described in the source text as the single most common error in trigonometry — the Law of Cosines is Pythagoras plus a correction term for the non-right angle."
          ],
          "hint": "Ask which law can start with the given data. One needs an angle-side pair that SAS does not provide."
        },
        {
          "id": "tls-014",
          "topic": "Law of Cosines and Pythagoras",
          "difficulty": "hard",
          "stem": "In the Law of Cosines c² = a² + b² − 2ab·cos C, what happens when C = 90°?",
          "choices": [
            "The formula becomes undefined.",
            "It reduces to the Pythagorean Theorem.",
            "It reduces to the Law of Sines.",
            "The correction term doubles."
          ],
          "correctIndex": 1,
          "explanations": [
            "Nothing becomes undefined. Cosine is perfectly well defined at 90°; it simply equals zero there.",
            "Correct. cos 90° = 0, so the term −2ab·cos C vanishes and the formula collapses to c² = a² + b². This is why the Law of Cosines is best understood as Pythagoras with a correction term measuring how far angle C departs from a right angle.",
            "The Law of Sines is a different relationship entirely, built on ratios of sines to opposite sides. No substitution turns one law into the other.",
            "The correction term disappears rather than doubling. It is largest when C is far from 90° and shrinks to nothing as C approaches it."
          ],
          "hint": "Evaluate the cosine at 90° and see what happens to the last term."
        },
        {
          "id": "tls-015",
          "topic": "The Ambiguous Case",
          "difficulty": "hard",
          "stem": "Why is SSA called the ambiguous case when using the Law of Sines?",
          "choices": [
            "The Law of Sines cannot be applied to SSA at all.",
            "The given data may correspond to two different triangles.",
            "The answer is always a non-real number.",
            "The triangle is always obtuse."
          ],
          "correctIndex": 1,
          "explanations": [
            "The Law of Sines can be applied to SSA — an angle is paired with its opposite side, which is exactly what it needs. The difficulty is with interpreting the result, not with starting.",
            "Correct. With two sides and a non-included angle, the side opposite the known angle can sometimes reach the base line in two places, producing two valid triangles — one acute and one obtuse at that vertex. This is the same ambiguity that disqualifies SSA as a congruence criterion.",
            "The computation returns an ordinary real sine value. The problem is that arcsine returns only one of the two angles that share that sine, and the supplementary angle may also be valid.",
            "Neither outcome is guaranteed. The ambiguity is precisely that BOTH an acute and an obtuse solution can exist for the same data — sometimes one, sometimes two, sometimes none."
          ],
          "hint": "Connect this back to why SSA fails as a congruence criterion. The same geometric freedom causes both problems."
        },
        {
          "id": "tls-016",
          "topic": "Trigonometric Area",
          "difficulty": "medium",
          "stem": "A triangle has sides a = 6 and b = 10 with an included angle C = 30°. Find its area.",
          "choices": [
            "15",
            "30",
            "60",
            "7.5"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The trigonometric area formula is A = ½ab·sin C. Here A = ½(6)(10)(sin 30°) = ½(60)(0.5) = 15.",
            "30 is ½ab without the sine factor, which would be the area only if C were 90° (since sin 90° = 1). The angle here is 30°, so the sine factor halves it again.",
            "60 is the raw product ab, with neither the ½ nor the sine applied.",
            "7.5 applies the ½ twice, or uses sin 30° = 0.25. The sine of 30° is exactly 0.5."
          ],
          "hint": "The formula needs the angle BETWEEN the two sides. Recall the exact value of sin 30° rather than approximating."
        },
        {
          "id": "tls-017",
          "topic": "Law of Sines Setup",
          "difficulty": "medium",
          "stem": "In a triangle, angle A = 40°, angle B = 60°, and side a = 8. Which equation correctly finds side b?",
          "choices": [
            "b = 8 · sin 40° / sin 60°",
            "b = 8 · sin 60° / sin 40°",
            "b² = 8² + b² − 2(8)(b)cos 60°",
            "b = 8 · cos 60° / cos 40°"
          ],
          "correctIndex": 1,
          "explanations": [
            "This inverts the ratio. Since angle B (60°) is larger than angle A (40°), side b must be LONGER than side a — but this expression gives a value smaller than 8.",
            "Correct. The Law of Sines gives sin A / a = sin B / b, so b = a · sin B / sin A = 8 · sin 60° / sin 40°. Because B > A, the result exceeds 8, as the larger-angle-opposite-larger-side rule requires.",
            "This is a malformed Law of Cosines with b appearing on both sides. In any case AAS data is exactly what the Law of Sines handles directly.",
            "The Law of Sines uses sines, not cosines. Substituting cosine changes the relationship entirely and will not produce a correct length."
          ],
          "hint": "Set up the proportion first, then sanity-check the answer against which angle is larger."
        },
        {
          "id": "tls-018",
          "topic": "Reasoning: Choose the Law",
          "difficulty": "hard",
          "stem": "You know all three sides of a triangle and need an angle. Which law applies, and why not the other?",
          "choices": [
            "Law of Sines, because it relates sides and angles.",
            "Law of Cosines, because it is the only one that can start from three sides (SSS).",
            "Either works equally well.",
            "Neither — three sides do not determine the angles."
          ],
          "correctIndex": 1,
          "explanations": [
            "The Law of Sines needs an angle to already be known, since every one of its ratios pairs a side with its opposite ANGLE. With only sides, there is nothing to start from.",
            "Correct. Rearranged, the Law of Cosines gives cos C = (a² + b² − c²)/(2ab), which needs only the three sides. Once one angle is known, the Law of Sines can finish the job.",
            "The Law of Sines cannot begin from SSS at all, so they are not interchangeable here.",
            "SSS determines a triangle completely — that is exactly why SSS is a congruence criterion."
          ],
          "hint": "Write out what each law needs before it can be used. Which one can start with no angle at all?",
          "kind": "reasoning"
        },
        {
          "id": "tls-019",
          "topic": "Reasoning: The Ambiguous Case",
          "difficulty": "hard",
          "stem": "Using the Law of Sines with SSA data, a student finds sin B = 0.6 and reports B = 36.87° as the only answer. What has been overlooked?",
          "choices": [
            "Nothing; sine gives a unique angle.",
            "The obtuse solution: sin(180° − 36.87°) = 0.6 as well, so B ≈ 143.13° may also produce a valid triangle.",
            "The angle should have been found with cosine instead.",
            "Sine values above 0.5 have no solution."
          ],
          "correctIndex": 1,
          "explanations": [
            "Sine is positive in both the first and second quadrants, so an equation like sin B = 0.6 has two solutions between 0° and 180°.",
            "Correct. This is the ambiguous case of SSA. Both 36.87° and 143.13° have a sine of 0.6, and each must be tested to see whether the angles still sum to under 180°.",
            "Cosine would avoid the ambiguity, but the Law of Sines was the right tool for the data — the fix is to check the second solution, not change law.",
            "Sine takes every value from 0 to 1 for angles in this range."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}