Ratios and Graphs of Trigonometric Functions
How to Sketch a trigonometric graph
Use this to sketch the graph of sine, cosine or tangent over a given range.
Before you start
- The shape of the basic sine, cosine and tangent curves
- Reading angles in degrees along an axis
- The idea of period and amplitude
When to use it
Use this to sketch the graph of sine, cosine or tangent over a given range.
The steps
- Note the shape of the basic curve for sine, cosine or tangent.
- Mark the key points: maximum, minimum and where it crosses the axis.
- Use the range in the question to decide how many cycles to draw.
- Plot the key points and join them with a smooth curve.
- Label the axes and the important values.
Worked example
Sketch the graph of y = cos x for 0° ≤ x ≤ 360°.
- The cosine curve is a smooth wave that starts at its maximum when x = 0°.
- Key points: maxima (0°, 1) and (360°, 1); minimum (180°, −1); axis crossings at (90°, 0) and (270°, 0).
- From 0° to 360° is exactly one full cycle, so draw one complete wave.
- Plot (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1) and join with a smooth curve.
- Label the x-axis in degrees and the y-axis from −1 to 1, marking the key values.
A second example, with a twist
The angle is 2x, which halves the period, so two complete cycles fit into 0° to 360°. Sketch the graph of y = sin 2x for 0° ≤ x ≤ 360°.
- The basic sine curve starts at zero and rises first; the 2x squeezes it horizontally.
- Period = 360° ÷ 2 = 180°, so one cycle ends at x = 180°: maximum (45°, 1), zero (90°, 0), minimum (135°, −1), zero (180°, 0).
- The range 0° to 360° is two periods, so draw two complete cycles.
- Plot the first cycle's key points, then repeat the same pattern from 180° to 360° (maximum at 225°, minimum at 315°).
- Label the x-axis in degrees and the y-axis from −1 to 1, marking 45°, 90°, 135°, 180° and so on.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
How do I work out how many cycles to draw?
Divide the range by the period. Basic sine and cosine have a period of 360°, so a 0° to 360° range holds one cycle.
When the angle is nx the period becomes 360° ÷ n, so y = sin 2x has a period of 180° and fits two cycles into the same range.
Where does the sine graph start compared with the cosine graph?
Sine starts at zero and rises, reaching its maximum of 1 at 90°; cosine starts at its maximum of 1 when x = 0°. Getting this first point right is the quickest way to keep the whole sketch correct, so always fix the starting point before you draw the rest of the curve.
Does the tangent graph look like sine and cosine?
No. Tangent is not a wave that stays between −1 and 1.
It rises steeply, is undefined at 90° and 270° where it has vertical asymptotes, and repeats every 180°. Draw the broken vertical asymptote lines first, then the rising branches between them.