IGCSE Maths Trigonometric Graphs: Sine and Cosine
Sine and cosine graphs model repeating patterns. Unlike a straight line, their values rise and fall in cycles, and both stay between −1 and 1 before vertical scaling. IGCSE questions may involve sketching standard curves, identifying maxima and zeros, or interpreting transformations. Understanding a few key coordinate values is a reliable starting point.
Plot the basic sine graph
For y = sin x in degrees, useful values over one full cycle are sin 0° = 0, sin 90° = 1, sin 180° = 0, sin 270° = −1 and sin 360° = 0. Plot these points and join them with a smooth wave, not straight segments. The function repeats after 360°, which is its period. Mark the angle scale clearly and use a calculator in the correct mode if you check intermediate values.
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Plot the cosine graph
For y = cos x, the corresponding values are 1, 0, −1, 0 and 1 at 0°, 90°, 180°, 270° and 360°. Cosine begins at its maximum at x = 0°, unlike sine. It also has period 360° and amplitude 1. Both graphs are smooth, periodic and bounded. Comparing the two side by side helps students recognise that they have the same shape but a horizontal phase difference.
Understand amplitude
In y = 2 sin x, all y-values are twice those of sin x, so the amplitude is 2. The maximum is 2 and the minimum is −2. For y = −3 cos x, amplitude is 3 but the negative sign reflects the graph in the horizontal axis. Do not claim an amplitude of −3; amplitude is a nonnegative distance from the centre line to a peak. Keep vertical scaling separate from horizontal stretching.
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Understand period changes
The function y = sin(2x) completes two cycles in 360°, so its period is 180°. More generally, for y = sin(bx) in degrees with positive b, the period is 360°/b. The same principle applies to cosine. Ensure multiplication is inside the sine or cosine argument: sin(2x) is not the same as 2 sin x. One changes the period, while the other changes the amplitude.
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Read horizontal shifts carefully
For y = cos(x − 30°), the usual cosine curve is shifted 30° to the right. A useful feature is the maximum at x = 30°, because the cosine argument is then zero. Although the expression contains a subtraction, the graph shifts right, not left. Check the position of a known peak or zero to reason about the transformation instead of relying only on a memorised sign rule.
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Avoid common sketching mistakes
Mark axes and scales before plotting. A sine or cosine curve should be smooth and periodic, not a series of sharp triangles. Keep degree and radian conventions distinct and follow the exam question’s stated units. If asked for intercepts in a restricted interval, report only those inside it. Evaluate a few known angles to check amplitude, period and phase before finalising the sketch.
Further reading: GCSE Maths Circle Theorems: Common Exam Traps
Frequently asked questions
What is the period of sin x in degrees?
The standard sine graph repeats every 360 degrees.
Does 2 sin x have the same period as sin x?
Yes. Its amplitude doubles, but its period remains 360 degrees.
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