Sine, Cosine and Area Rules

Sine, Cosine and Area Rules

Use the sine rule when you know an angle-opposite-side pair; use the cosine rule when you have SAS or SSS information; use the area rule $A=\tfrac12 ab\sin C$ for two sides and their included angle. Sketch and label the triangle first. Calculator mode must be degrees unless the question explicitly uses radians.

Worked reasoning

With sides 7 and 10 enclosing $60^\circ$, area $=\tfrac12(7)(10)\sin60^\circ=35(0.866...)\approx30.3$ square units.

Exam method

1. Label opposite pairs and the included angle. 2. Select the rule from the given information. 3. Substitute with units, round only at the end, and test reasonableness.

Choose a rule from a data map

Make a quick inventory before writing a formula: do you have an opposite side-angle pair, two sides with the included angle, or all three sides? The sine rule has an ambiguous-case possibility, so a diagram and a check for a second triangle matter when SSA data arise. Carry full calculator precision through intermediate steps and round only the stated final answer. This prevents a small early rounding change from turning into a mark-losing angle error.

One-minute retrieval: Sine, Cosine and Area Rules

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.