IGCSE Maths / Graphs and transformations
Sine, cosine and tangent graphs revision notes
Edexcel International GCSE Maths (4MA1), chapter 26: Graphs and transformations, specification point 3.3a. Free lesson.
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The graphs of sin x, cos x and tan x have different starting points, repeats and ranges. Knowing their key points helps you sketch them and find every solution in a given angle range.
How to sketch y = sin x
Sine makes a repeating wave. Plot its key points first, then join them with a smooth curve. It starts at the origin, rises to its maximum, then falls through the axis and reaches its minimum. At 0°, sin x = 0, so the graph starts at (0, 0). At 90°, sin x = 1, the highest value. At 180°, sin x = 0, so the curve crosses the axis. At 270°, sin x = −1, the lowest value. At 360°, sin x = 0 again. The wave repeats every 360°, and continues for angles beyond this range.
How to sketch y = cos x
Cosine has the same wave shape as sine, but it begins a quarter-turn earlier. This makes its starting point the main feature to remember. At 0°, cos x = 1, so the graph starts at (0, 1). At 90°, cos x = 0, where the curve first crosses the axis. At 180°, cos x = −1, its lowest point. The curve rises back to 1 at 360°. Like sine, cosine repeats every 360°. You can get the cosine graph by shifting the sine graph 90° to the left.
How to sketch y = tan x
The tangent graph is different from the two waves. It has separate rising branches, with breaks where the curve gets closer and closer to a vertical line but never reaches it. Such a line is called an asymptote. The graph has breaks at 90° and 270°. These asymptotes are 180° apart. tan x = 0 at 0°, 180° and 360°. The graph repeats every 180° and has no upper or lower limit.
Compare the three graphs
Use the starting points, repeat intervals and ranges to check that a sketch has the right shape. The range means the possible values of y. Sine starts at (0, 0), repeats every 360° and has values from −1 to 1. Cosine starts at (0, 1), repeats every 360° and also has values from −1 to 1. Tangent starts at (0, 0), repeats every 180° and has no limit on its values. Since sine and cosine cannot go above 1 or below −1, sin x = 2 has no solutions.
Use a graph to find every solution
A calculator’s inverse sine function gives just one angle. A sketch shows whether the equation has other solutions in the required range. Draw the horizontal line for the given value of y, then read off every crossing. For 0° ≤ x ≤ 360°, the line y = 0.5 crosses y = sin x at 30° and 150°. For 0° ≤ x ≤ 360°, the line y = 0.5 crosses y = cos x at 60° and 300°.