To solve a trigonometric equation over an interval, find the acute reference angle, place it in every quadrant where the function has the right sign, and keep only the angles inside the interval.
This lesson is part of trigonometric functions. Some equations first need an identity from using identities to simplify expressions.
What is the quadrant method?
Follow the same four steps each time.
- Isolate the function, for example sin x = 0.5.
- Find the acute reference angle, using the positive value.
- Decide the quadrants from the sign: sine is positive in quadrants 1 and 2, cosine in 1 and 4, and tangent in 1 and 3.
- Write each angle and keep those inside the interval.
Worked example 1: a single angle
Solve 2 sin x = 1 for 0° ≤ x ≤ 360°.
sin x = 0.5, and the reference angle is 30°. Sine is positive in quadrants 1 and 2, so x = 30° or x = 180° − 30° = 150°.
Worked example 2: an equation needing an identity
Solve 2 sin²x = 3 cos x for 0° ≤ x ≤ 360°.
Replace sin²x with 1 − cos²x: 2(1 − cos²x) = 3cos x, so 2cos²x + 3cos x − 2 = 0. Factorise: (2cos x − 1)(cos x + 2) = 0.
So cos x = ½ or cos x = −2. The value −2 is outside the range of cosine, so it is rejected. With cos x = ½, the reference angle is 60°, and cosine is positive in quadrants 1 and 4: x = 60° or x = 300°.
Worked example 3: a multiple angle
Solve sin 2x = 0.5 for 0° ≤ x ≤ 360°.
Stretch the interval: if 0° ≤ x ≤ 360°, then 0° ≤ 2x ≤ 720°. Solve for 2x: the reference angle is 30°, so 2x = 30°, 150°, 390°, 510° (adding 360° to the first two).
Divide by 2: x = 15°, 75°, 195°, 255°. All four lie inside the original interval.
The mistake that costs marks
The common slip is to stop at the calculator answer. In example 3, a student writes 2x = 30°, so x = 15°, and misses the other three solutions.
| Step | Wrong | Right |
|---|---|---|
| Interval for 2x | 0° to 360° | 0° to 720° |
| Angles for 2x | 30°, 150° | 30°, 150°, 390°, 510° |
| Answers for x | 15°, 75° | 15°, 75°, 195°, 255° |
Write the stretched interval beside the equation before you solve. It reminds you to list more angles.
A radian example
Solve tan x = 1 for 0 ≤ x ≤ 2π. The reference angle is π ÷ 4, and tangent is positive in quadrants 1 and 3, so x = π ÷ 4 or 5π ÷ 4.
Check yourself
Solve cos 2x = 0.5 for 0° ≤ x ≤ 180°.
Answer
The interval for 2x is 0° ≤ 2x ≤ 360°. Cos 2x = 0.5 has reference angle 60°, and cosine is positive in quadrants 1 and 4, so 2x = 60° or 300°.
Dividing by 2 gives x = 30° or 150°. Check: cos 60° = 0.5 and cos 300° = 0.5.
What to study next
Some equations mix sin 2x with sin x or cos x. Go on to using addition and double-angle formulas, where equations such as sin 2x = sin x become factorisable.
If you want a teacher to work through interval questions with you, see online one-to-one Additional Mathematics tuition.