To solve an equation such as sin x = 0.6 for 0° ≤ x ≤ 360°, draw the graph and a horizontal line at 0.6. Each place the line meets the curve is an answer.
This lesson follows reading sine, cosine and tangent graphs and belongs to the trigonometric ratios and graphs chapter.
How do I find every angle in the range?
Use the calculator for the first angle, then use the graph’s symmetry for the second. The pattern depends on the curve.
| Equation type | Calculator angle α | Other angle in 0° to 360° |
|---|---|---|
| sin x = positive | sin⁻¹ value | 180° − α |
| cos x = positive | cos⁻¹ value | 360° − α |
| tan x = positive | tan⁻¹ value | 180° + α |
Worked example: sin x = 0.6
Draw y = sin x and the line y = 0.6. The line crosses the curve twice between 0° and 360°, once on the way up and once on the way down.
- Calculator: α = sin⁻¹ 0.6 = 36.87°, so the first angle is 36.9°.
- The curve is symmetrical about 90°, so the second angle is 180° − 36.87° = 143.1°.
- Check: sin 143.1° = sin 36.9° = 0.6.
Now try cos x = 0.3. The calculator gives 72.54°. The cosine curve is symmetrical about 180°, so the second angle is 360° − 72.54° = 287.46°. The answers are 72.5° and 287.5°.
The mistake that costs marks
The common slip is to write only the calculator angle. For tan x = 2, a student writes x = 63.4° and stops.
| Step | Wrong | Right |
|---|---|---|
| Calculator | 63.4° | 63.4° |
| Count crossings | (not done) | tangent repeats every 180°, so two in the range |
| Second angle | missing | 180° + 63.4° = 243.4° |
| Answer | 63.4° | 63.4° and 243.4° |
When a question says “find all values of x”, the marks usually include the second angle.
Check yourself
Solve sin x = −0.5 for 0° ≤ x ≤ 360°.
Answer
Find the reference angle from the positive value: sin⁻¹ 0.5 = 30°.
Sine is negative in quadrants 3 and 4. So x = 180° + 30° = 210° and x = 360° − 30° = 330°.
Check on the graph: the line y = −0.5 crosses the curve at 210° and 330°, below the axis.
What to study next
Next, see how stretching a graph changes its height and its period in interpreting amplitude, period and simple transformations. Then test the whole chapter with the trigonometry practice set, and record slips in the mistake log and paper-error review.
For a teacher to check your working on these equations, see online one-to-one Mathematics tuition.