lesson
Match each side with its opposite angle
1 · Learn
The sine rule compares a/sin A with b/sin B, using opposite pairs. The cosine rule a² = b² + c² - 2bc cos A uses the included angle A opposite side a. These extend calculation beyond right triangles.
Choose a rule suited to the known information. Some sine-rule configurations can have two possible triangles; do not discard ambiguity without checking conditions.
2 · Worked example
With sides 5 and 7 enclosing 60°, the opposite side has square 25 + 49 - 70 × 0.5 = 39, so its length is √39.
3 · Your turn
Two sides of length 4 enclose 60°. Find the opposite side using the cosine rule.
Check your answer
Its square is 16 + 16 - 32 × 0.5 = 16, so the side is 4.
4 · Apply your learning
Label opposite pairs on a non-right triangle and justify which rule uses the supplied information.
Adult guidance and safety
This sequence includes extension mathematics and assumes secure earlier work. Check prerequisite understanding and the learner's readiness before assigning extension tasks. Require derivations, accurate diagrams and varied practice; a correct short example is not full assessment evidence. Exam-board and tier-specific pathways are outside this edition.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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