Measurement applications: cosine rule

Task goals

• Apply the cosine rule (a^2=b^2+c^2-2bc cos A) to find a missing side of a non-right-angled triangle in a real-world measurement scenario, given two sides and the angle between them (SAS).
• Correctly identify the INCLUDED angle between two given sides (the angle at the vertex where both known sides meet), and recognise this is the only angle usable in the SAS side-finding form.
• Apply the rearranged cosine rule (cos A=(b^2+c^2-a^2)/(2bc)) to find a missing angle of a non-right-angled triangle, given all three sides (SSS).
• Rearrange consistently for the angle being solved -- if solving for angle A, correctly use side a (opposite A) and the other two sides b and c.
• Recognise that a negative value inside inverse cosine signals a valid obtuse angle between 90 and 180 degrees, not an error, and compute it correctly.
• Check that three given side lengths satisfy the triangle inequality before attempting to solve for an angle.
• Use a length or angle found with the cosine rule as an input to a further measurement, cost, time, or compliance calculation, never leaving a bare "find x" answer unused.
• Convert between units of length before or after applying the cosine rule, and round an irrational answer to a stated number of decimal places or degrees without losing precision partway through a multi-step calculation.
• Check a calculated angle or side against a legal minimum/maximum, a stability limit, or a required frontage, and state whether a real-world structure or subdivision complies.
• Combine a cosine-rule side-find (SAS) with a second cosine-rule angle-find (SSS) and the angle sum, and where genuinely useful a SOHCAHTOA height decomposition cross-checked two ways, within the same problem.
• Compare two real material, route, or method options using cosine-rule-derived numbers, check a stated budget, and write a short, numbers-based justification for a real-world measurement decision.