Mathematics 0580 · Syllabus tracker
Trigonometric Graphs
Exact trig values, the graphs of y = sin x, cos x and tan x from 0° to 360°, and solving trig equations.
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Quick revision
Trigonometric Graphs on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Exact trig values, the graphs of y = sin x, cos x and tan x, and solving trig equations from 0° to 360°.
1Exact values
- Learn the table — it is needed on the non-calculator paper.
- Memory aid: sin goes 0, ½, √2/2, √3/2, 1 from 0° to 90°; cos is the same list backwards.
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | ½ | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | ½ | 0 |
| tan | 0 | √3/3 | 1 | √3 | undefined |
Worked example
Work out 4 sin 30° + tan 45°.
- sin 30° = ½ and tan 45° = 1.
- 4 × ½ + 1.
3
2The graphs of sin, cos and tan
- y = sin x: starts at 0, maximum 1 at 90°, 0 at 180°, minimum −1 at 270°, 0 at 360°.
- y = cos x: starts at 1, 0 at 90°, −1 at 180°, 0 at 270°, 1 at 360°.
- y = tan x: through (0, 0), (180°, 0), (360°, 0); asymptotes at x = 90° and 270°; repeats every 180°.
- y = a sin x stretches it vertically: maximum a, minimum −a.
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
| cos x | 1 | 0 | −1 | 0 | 1 |
| tan x | 0 | — | 0 | — | 0 |
Worked example
State the maximum and minimum values of y = 3 cos x and where the maximum occurs (0° ⩽ x ⩽ 360°).
- cos x is between −1 and 1.
Maximum 3 at x = 0° and 360°; minimum −3
3Solving trig equations
- Rearrange to sin x = k, cos x = k or tan x = k first.
- The calculator gives one angle; use the graph's symmetry for the others in 0° to 360°.
How to do it
- Rearrange to trig function = number.
- Use the inverse on the calculator to get x₁.
- Find the second solution from the table (add 360° to any negative angle first).
- Check both are in the range.
| Equation | Solutions in 0°–360° (x₁ from the calculator) |
|---|---|
| sin x = k | x₁ and 180° − x₁ |
| cos x = k | x₁ and 360° − x₁ |
| tan x = k | x₁ and x₁ + 180° |
Worked example
Solve 2 sin x + 1 = 0 for 0° ⩽ x ⩽ 360°.
- sin x = −0.5.
- Calculator: x = −30°, outside the range.
- Solutions: 180° + 30° = 210° and 360° − 30° = 330°.
x = 210° or x = 330°
Exam tip: Sketch the graph with the horizontal line y = k to see how many solutions there are.
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Giving only the calculator's answer | There are usually two answers in 0°–360° |
| sin x = −0.5 → x = −30° | Negative angles are outside the range: use 210° and 330° |
| Calculator in radians | Set it to degrees |
| tan x = k → x₁ and 180° − x₁ | tan repeats every 180°: x₁ and x₁ + 180° |
★In the exam
- Learn the exact values — they appear on the non-calculator paper.
- Sketch the graph to check your second solution.
- Give angles to 1 decimal place unless they are exact.