Measurement applications: SOHCAHTOA

Task goals

• Apply SOHCAHTOA (sin, cos, tan) to find a missing side of a right-angled triangle in a real-world measurement scenario, given an angle and one side.
• Apply the inverse trigonometric functions (sin^-1, cos^-1, tan^-1) to find a missing angle of a right-angled triangle, given two sides.
• Correctly identify which side is opposite and which is adjacent relative to the GIVEN or MARKED angle, recognising that this changes depending on which acute angle is used.
• Decide, from a real-world description or an unlabelled diagram, which trigonometric ratio applies, rather than being told directly.
• Use a length or angle found with a trigonometric ratio as an input to a further measurement, cost, or compliance calculation, never leaving a bare "find x" answer unused.
• Convert between units of length before or after applying a trigonometric ratio, 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 against a legal maximum gradient or a manufacturer's safe-use range, and state whether a real-world structure complies.
• Use a trig-derived length, a material's coverage rate or unit size, and a price to find a total quantity and cost.
• Combine a trigonometric step with Pythagoras' theorem within the same problem when genuinely useful, such as cross-checking a hypotenuse.
• Compare two real material, supplier, or design options using trig-derived numbers, check a stated budget or code requirement, and write a short, numbers-based justification for a real-world measurement decision.