I keep tripping over the sine rule when I’m solving for an angle. For example, I had a triangle with A = 30°, a = 7 (opposite A), and b = 10. Using the sine rule I got sin B = 10 * sin 30° / 7 ≈ 0.714…, so my calculator spits out B ≈ 45.6°. But then my teacher said B could also be 180° − 45.6° ≈ 134.4°, and now I’m second-guessing myself every time. I get especially lost when the diagram isn’t to scale, because I pick the wrong one and only realize later that the sides don’t make sense.
Is there a clean way to tell, just from the numbers, whether I should take the acute angle, the obtuse one, or consider both? Like a quick checklist for when there are 0, 1, or 2 possible triangles with the sine rule (that “ambiguous case” thing). And if both angles seem possible, how do you decide which one actually matches the given triangle without redrawing it ten times? I feel like I’m overthinking this!
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3 Responses
You’ve run into the classic SSA “ambiguous case,” where sin B determines a line of sight but not which side of the line you’re on. Here’s the clean test I use when A and the two opposite sides a (opposite A) and b (opposite B) are given. First compute h = b sin A (think of h as the “height” needed for side a to reach side b when you drop a perpendicular). If A is acute: (i) a < h → no triangle; (ii) a = h → exactly one right triangle (B = 90°); (iii) h < a < b → two triangles (B is either arcsin(b sin A / a) or 180° minus that); (iv) a ≥ b → exactly one triangle (B is the acute arcsin value). If A is obtuse: you can only have a triangle when a > b; otherwise none. A quick way to pick between the acute and obtuse options without redrawing is the “larger side ↔ larger angle” rule: if b ≤ a then B ≤ A, so only the acute arcsin value fits; if b > a and a > h, both angles fit and you should report both (unless some extra information rules one out). In your numbers A = 30°, a = 7, b = 10, so h = 10 sin 30° = 5 and we’re in h < a < b, hence two triangles: B ≈ 45.6° or 134.4°. Analogy: picture side a as a door of fixed length swinging from vertex A toward the line containing side b-too short (a < h) it can’t reach, just right (a = h) it taps the line at a right angle, a bit longer (h < a < b) it touches in two positions (two B’s), and once it’s long enough (a ≥ b) it hits in only one place.
Quick SSA checklist (angle A given, opposite side a, other side b): let h = b·sin A; if a < h → 0 triangles, a = h → 1 (right), h < a < b → 2 (B acute and obtuse), a ≥ b → 1 (B acute); if A ≥ 90°, then a > b gives 1 triangle, else 0-here h=5 with a=7https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-law-of-sines/v/ambiguous-case-of-law-of-sines; do you want me to compute C and c for both possibilities or show a quick sketch trick?
Here’s the no-drama way I keep the sine-rule “ambiguous case” straight: compute the height h = b·sin A. Then run the quick checklist: if a < h, no triangle; if a = h, exactly one right triangle; if h < a < b, two triangles (B could be acute or obtuse); if a ≥ b, one triangle and B is acute. In your example A = 30°, a = 7, b = 10, so h = 10·sin 30° = 5, and since 5 < 7 < 10 you’re in the two-triangles zone; sin B = 5/7 gives B ≈ 45.6° or 134.4°, and both are genuinely possible from just those numbers. To choose between them, don’t trust the sketch (those things lie) - use the size rule: larger side ↔ larger opposite angle. If you also know or can infer which angle in the picture is the “wide” one, pick the version of B that makes the side–angle ordering consistent; otherwise, you can’t break the tie without extra info, and that’s not you overthinking - that’s the data being insufficient. One more mental trick: if a ≥ b, angle A can’t be smaller than B, so B stays acute; if a = b, then A = B exactly. I’ll admit I used to botch this on speed drills until I started jotting h = b·sin A at the top of the page; after choosing the “pretty” obtuse angle once and getting a third side longer than both given sides, I learned my lesson. Slight caveat that sometimes when a is much larger than b, people expect B to be obtuse “by feel,” but with A acute that’s actually not how the math plays out - go by the checklist. If you want a clean walkthrough with pictures, Khan Academy’s ambiguous case page is solid: https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-ambiguous-case/a/ssa-ambiguous-case.