Reduction Formulae and Signs

Reduction Formulae and Signs

Reduction formulae turn angles outside the first quadrant into a reference angle. First locate the quadrant, decide the sign of the required ratio using CAST or a unit-circle sign chart, then use the reference angle. Exact values come from special angles after the reduction, not from guessing a sign.

Worked reasoning

$\sin150^\circ=\sin(180^\circ-30^\circ)=+\sin30^\circ=1/2$. It lies in quadrant II, where sine is positive.

Exam method

1. Express the angle using 90, 180, 270 or 360 degrees. 2. Identify the quadrant and sign. 3. Evaluate the reference angle exactly.

Build an exact-value reference set

Know the exact sine, cosine and tangent values for $0^\circ,30^\circ,45^\circ,60^\circ$ and $90^\circ$, but do not memorise them as isolated fractions. A 30-60-90 triangle and a 45-45-90 triangle explain each value. In a reduction question, write the reference angle separately from the original angle and then attach the quadrant sign. This separates magnitude from sign and makes calculator-free reasoning much safer.

One-minute retrieval: Reduction Formulae and Signs

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.