Trigonometric Graphs and Equations

Pythagoras and Trigonometry

What a sketch must show

A sketch is asked for in its own right, usually for 2 marks, and it needs the shape and the key points rather than plotted accuracy
Mark where the curve meets the axes, and show the maximum of 1 and minimum of −1 for sine and cosine
Run the curve across the whole interval the question states, no further and no less
Draw it freehand and smoothly, never with a ruler
For a tangent sketch, put the asymptotes in as dotted vertical lines, since the branches make no sense without them
Where the sketch is part (a) and an equation is part (b), the sketch is meant to be used, so draw the horizontal line on it and mark the crossings

Exam tip

Give every solution inside the stated interval, not just the calculator's. Sine's partner is 180° minus the first; cosine's is 360° minus the first, and mixing those two up is the standard error. The answer line usually shows two spaces joined by "or", which tells you how many are expected. Keep the calculator in degrees and give angles to 1 decimal place.

Worked example

Sketching a curve to place the answers

Sketch y = cos x for 0° ⩽ x ⩽ 360° and use it to explain why cos x = −0.7 has two solutions in that interval.

Solution:

Mark the axes first: x from 0° to 360°, and y from −1 to 1, since the curve never leaves that range
Plot the five points the curve is fixed by: (0, 1), (90, 0), (180, −1), (270, 0) and (360, 1)
Join them with one smooth curve, drawn freehand rather than ruled
Draw the horizontal line y = −0.7 across the sketch
The line sits below the axis, and the curve is below the axis between 90° and 270°
The line therefore crosses the curve twice, once on the way down and once on the way back up
So there are two solutions, one between 90° and 180° and one between 180° and 270°
The sketch is what shows where the second answer lives; the calculator only ever gives the first