Quarter-Cycle Points

What is Quarter-Cycle Points?

Quarter-cycle points are the five equally spaced x-values that determine one complete cycle of a sine or cosine graph: the start of the cycle, the three points a quarter, a half and three quarters of the way through it, and the end. They are spaced one quarter of a period apart, and at them the curve is always at a maximum, a minimum or a centre-line crossing, so plotting them and joining with a smooth curve reproduces the cycle exactly without any further calculation.

This definition is part of the Trigonometry chapter in Cambridge O Level Additional Mathematics.

Concepts related to Quarter-Cycle Points

• period
• centre line
• exact coordinates
• smooth curve

Common mistakes with Quarter-Cycle Points

• Quarter-cycle points plotted by eye, so the second half of the cycle is narrower than the first. Why it fails The four quarters of a period are equal by definition. Unequal spacing changes the period part-way through the sketch, which is a different function. Fix Compute \(\dfrac{\text{period}}{4}\) once, mark that interval repeatedly along the axis with a ruler, and plot only at those marks.