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Sine and Cosine Rules Questions
A question bank for senior trigonometry — known internationally as the sine rule and cosine rule (or the law of sines and law of cosines). Each question asks students to find an unknown side or angle in a non-right triangle, with coverage of SSS, SAS, the ambiguous case, and the area formula ½ab sin C.
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What's Included in the Sine and Cosine Rules Question Bank?
🔥Progressive Difficulty Levels
The bank works through every standard setup students meet in senior maths. Sine rule for opposite pairs (an angle and its opposite side, plus one more piece of information). Cosine rule for SAS (two sides and the included angle) and SSS (all three sides). A dedicated set on the ambiguous case where two triangles satisfy the given information, plus area-of-a-triangle questions using ½ab sin C.
🌍 Targeted Practice
Questions move from setup to application. Students start with straightforward triangle diagrams labelled A, B, C with sides a, b, c, then progress to bearings, surveying scenarios, and contextual word problems where the triangle has to be drawn before the rule is chosen. Worked answers show the rule selected and why — useful for teachers marking common errors like missing the ambiguous case.
💡Conceptual Understanding
Built for Year 10 through senior secondary — aligned to the Australian Curriculum Year 10 / Year 10A trigonometry strands and equally suited to Algebra 2 and Pre-Calculus units on non-right triangles. Difficulty is graded so the same topic supports students consolidating the basics and students preparing for VCE, HSC, ATAR, IB, SAT, or ACT trigonometry questions.
Practice Questions
Practice Questions
Mixed practice on sine rule, cosine rule, the ambiguous case, and triangle area — with full worked solutions so students can self-check or teachers can use them as exemplars.
Engaging Exercises
Engaging Exercises
Application questions covering bearings, surveying, navigation, and engineering contexts — the settings most often used in exam questions on non-right triangles.
Differentiated Questions
Differentiated Questions
Scaffolded entry questions for students still learning when to apply each rule, plus extension questions involving multi-step problems, proofs of the rules, and combined sine/cosine setups.
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What is covered in the sine and cosine rules question bank?
Choosing between the sine rule and the cosine rule
The biggest source of mistakes in this topic is rule selection. Questions in this set give students the information first and ask them to decide before solving. Sine rule when there's an opposite pair (a side and its opposite angle) plus one more piece. Cosine rule when the setup is SAS or SSS. Mixed sets keep students from defaulting to whichever rule was taught most recently.
The ambiguous case of the sine rule
The sine rule states that a/sin A = b/sin B = c/sin C for any triangle, and the cosine rule states that a² = b² + c² − 2bc cos A. The bank covers both directly and through rearrangement — finding a missing side, finding a missing angle, and using ½ab sin C to find triangle area without the perpendicular height. Worked solutions show the algebraic step where students most often drop a sign or use the wrong inverse function.
Bearings, surveying, and real-world non-right triangle problems
When two sides and a non-included angle are given (SSA), there can be two valid triangles, one triangle, or none. A dedicated set walks students through identifying which case applies, calculating both possible angles using the supplementary identity sin(180° − θ) = sin θ, and checking each answer against the triangle inequality. This is the highest-value section for senior students — ambiguous-case questions appear regularly in VCE, HSC, IB, and Pre-Calculus exams and are the most commonly missed.
Browse the full sine and cosine rules question bank and assign a set that matches your class today