This chapter extends trigonometry from right-angled triangles to any angle and to graphs. It has four lessons and a practice set, and sits inside the wider SPM Mathematics guide.
What does this section cover?
Start with using trigonometric ratios beyond acute angles, which explains reference angles and signs by quadrant. Then read sine, cosine and tangent graphs so the shapes become familiar.
The third lesson is finding angles from a trigonometric graph. The last is interpreting amplitude, period and simple transformations. The practice set mixes all four.
One value, two angles
Between 0° and 360°, each of these equations has two answers.
| Equation | Calculator gives | Other angle | Why |
|---|---|---|---|
| sin θ = 0.5 | 30° | 150° | Sine is positive in quadrants 1 and 2 |
| cos θ = 0.5 | 60° | 300° | Cosine is positive in quadrants 1 and 4 |
| tan θ = 1 | 45° | 225° | Tangent is positive in quadrants 1 and 3 |
The calculator always returns one angle. The graph or the quadrant rule supplies the other.
This is why the lessons run in the order above. Angles beyond 90° give you the signs, the graphs show those signs as a picture, and the last lesson changes the picture by stretching it.
Who should start where?
If sin 150° or cos 240° makes you pause, begin with angles beyond 90°. If you can calculate but the graph pictures are blurry, begin with reading the three graphs.
If you find one angle but the question asks for all angles, begin with finding angles from a graph. If transformed graphs such as y = 3 sin 2x look confusing, begin with amplitude and period.
For a teacher to go through your working question by question, see online one-to-one Mathematics tuition.