Ratios and Graphs of Trigonometric Functions

You already know sine, cosine and tangent for angles in a right-angled triangle. This chapter extends them to any angle from 0° to 360°, teaches you to find their values (including negative ones), and to draw and read the wave-like graphs of these functions.

Paper 2 often asks you to sketch a sine or cosine graph over a stated range, then use it to find angles that satisfy an equation. The graph must be neat and correctly scaled; the sign of a ratio in the second, third or fourth quadrant is where marks quietly go.

Every value in this chapter can be found with one reliable routine, so you never have to guess. First find the acute reference angle, the angle the arm makes with the horizontal x-axis: it is θ itself in the first quadrant, 180° − θ in the second, θ − 180° in the third, and 360° − θ in the fourth.

Take the ordinary sine, cosine or tangent of that acute angle, then attach a sign using the quadrant rule: going anticlockwise from the first quadrant, All, Sine, Tangent and Cosine are the ratios that stay positive in quadrants one, two, three and four. So cos 150° has reference angle 30° and sits in the second quadrant where cosine is negative, giving −cos 30°; sin 210° is −sin 30°; and tan 300° is −tan 60°.

Reference angle for the size, quadrant for the sign, that two-step habit removes almost every sign error.

A handful of values turn graph-sketching and equation-solving from slow to instant. Know the quadrantal points cold: sin 0° = 0, sin 90° = 1, sin 180° = 0, sin 270° = −1, sin 360° = 0, while cosine runs 1, 0, −1, 0, 1 across the same angles, these are the peaks, troughs and crossings your graph has to go through.

Know the three special acute angles as exact surds: sin 30° = ½ and cos 30° = √3/2; sin 45° = cos 45° = √2/2; sin 60° = √3/2 and cos 60° = ½; and tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Finally, remember that tangent is undefined at 90° and 270°, which is exactly where its graph has a vertical break instead of a smooth curve.

With these anchor values in your head you can place every important point of a sine, cosine or tangent graph without a calculator.

When a question says solve for 0° ≤ x ≤ 360°, it is almost always warning you that there is more than one answer, because a horizontal line at a given value cuts the sine or cosine curve twice in a full turn. Work in three steps.

First, use the positive value to find the basic acute angle from a calculator, ignoring the sign for now, for sin x = −0.5 you find sin⁻¹(0.5) = 30°. Second, decide which two quadrants the sign points to: a negative sine belongs to the third and fourth quadrants.

Third, build both answers from the reference angle: third quadrant gives 180° + 30° = 210°, fourth gives 360° − 30° = 330°. If the range were wider, say up to 720°, you would simply add 360° to each answer to collect the extra solutions.

Training yourself to expect two answers, then checking each lies inside the stated range, is what separates full marks from half.

This two-part example is the standard Paper 2 pairing, find a ratio in a non-first quadrant, then solve an equation across the full range.