The graphs
- y = sin x: starts at 0, reaches 1 at 90°, 0 at 180°, −1 at 270° and 0 at 360°. It repeats every 360°.
- y = cos x: starts at 1, is 0 at 90°, −1 at 180°, 0 at 270° and 1 at 360°. It repeats every 360°.
- y = tan x: 0 at 0°, 180° and 360°. It has asymptotes (where it is undefined) at 90° and 270°, and repeats every 180°.
- sin x and cos x always lie between −1 and 1.
Solving equations for 0° to 360°
- Use your calculator for the first solution, then use symmetry for the others.
- sin x = k: the solutions are x and 180° − x. sin x = 0.5 gives 30° and 150°.
- cos x = k: the solutions are x and 360° − x. cos x = 0.5 gives 60° and 300°.
- tan x = k: the solutions are x and x + 180°. tan x = 1 gives 45° and 225°.
Negative values
- For negative values, sketch the graph. sin x = −0.5: the calculator gives −30°, which isn't in range, so the solutions are 210° and 330°.
- 2cos x + 1 = 0 means cos x = −0.5, so x = 120° and 240°.
- 5cos x + 2 = 0 means cos x = −0.4, so x = 113.6° and 360° − 113.6° = 246.4° (to 1 decimal place).
Key terms
- Period
- The interval after which a graph repeats.
- Asymptote
- A line a graph approaches but never touches.
- Amplitude
- The height of a wave from its centre line.
- Principal value
- The first solution given by a calculator.
- Symmetry
- Using the shape of the graph to find other solutions.