A trigonometric problem may ask for a value that can be found without solving the angle. Look for an identity connecting the supplied expression directly to the requested one.
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Expand the given relationship
Suppose sinθ + cosθ = 1/2 for a real angle. Squaring gives sin2θ + cos2θ + 2sinθcosθ = 1/4.
The first two terms sum to 1, so 2sinθcosθ = -3/4. Therefore sin(2θ) = -3/4.
Check that the value is possible
The result lies between -1 and 1, as a sine value must. The original sum 1/2 also lies within its possible range from -√2 to √2.
These checks test plausibility; the identity provides the actual derivation.
Avoid claiming more than is given
The information does not automatically specify a unique angle or a unique sign for cos(2θ). Different angles can share the same given sum while differing in other expressions.
When solving for angles after squaring, return to the original unsquared condition to reject extraneous possibilities. This original example supports AMC 12 algebraic reasoning; current competition information should be checked through official MAA resources.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
