Common Misconceptions

⚠️ Common Misconceptions

Misconception 1: "The period of y = sin(2x) is 720° (twice 360°)"

When the angle is multiplied by a factor n inside the function, the period DIVIDES by n, it does not multiply. y = sin(2x) completes two full cycles in 360°, so its period is 360° ÷ 2 = 180°. The larger n is, the SHORTER and more compressed the graph becomes. Think of it as squashing the graph horizontally — more cycles fit in the same space. Only y = sin(x/2) would have a longer period of 720°.

Misconception 2: "sin x = 0.5 has only one solution between 0° and 360°"

The sine graph crosses any horizontal line y = k (where −1 < k < 1) TWICE in every 360° period. For sin x = 0.5, the solutions are x = 30° AND x = 150°. The sine graph is symmetric about x = 90°, so the second solution is always 180° − (first solution). For cosine, the graph is symmetric about 0° and 360°, so the second solution is 360° − (first solution). Always check for two solutions in trig equations.

Misconception 3: "y = cos x starts at zero like y = sin x"

y = cos x starts at its MAXIMUM value of 1 when x = 0, not at zero. This is because cos 0° = 1 (the cosine of zero degrees is 1). The cosine graph looks like the sine graph shifted 90° to the left — it starts at a peak rather than at the middle. If you mistakenly sketch cos as starting at zero, all your key points will be wrong. Always mark (0°, 1) as the starting point for the cosine graph.