MISSION · Geometry
Solving Trigonometric Equations
Why the calculator gives one answer when there are many
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LEVEL 2 · LEARN
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The idea in one sentence
Trigonometric functions repeat, so their equations have infinitely many solutions — and the calculator hands you exactly one of them.
The calculator’s blind spot
Solve \(\sin x = 0.5\) for \(0° \le x \le 360°\).
Your calculator gives \(x = 30°\). That’s correct, and incomplete: \(\sin 150°\) is also 0.5. Two solutions in the range, and the calculator mentioned one.
It isn’t malfunctioning. \(\sin^{-1}\) has to be a function, so it must return a single value, and the convention is to return the one between \(-90°\) and \(90°\). That returned value is called the principal value, and finding the rest is your job.
Finding the others
The unit circle shows where they are. Sine is the height of the point, so any horizontal line across the circle crosses it twice — once on the right half, once on the left.
- \(\sin x = k\): the second solution is \(180° - x\).
- \(\cos x = k\): the second is \(360° - x\), since cosine is symmetric about the horizontal axis.
- \(\tan x = k\): solutions repeat every \(180°\), so add \(180°\).
Then add or subtract full turns of \(360°\) (\(180°\) for tangent) to reach every solution in whatever range the question gives.
For \(\sin x = 0.5\) over \(0°\) to \(720°\): \(30°\), \(150°\), \(390°\), and \(510°\) — four of them.
Read the range first
The range is part of the question, and forgetting it produces either too few solutions or too many. Check every candidate against the stated interval before writing it down, and note whether the endpoints are included.
When the angle is multiplied
\(\sin 2x = 0.5\) for \(0° \le x \le 360°\) needs care.
If \(x\) runs to \(360°\), then \(2x\) runs to \(720°\) — so solve for \(2x\) over the doubled range, finding \(30°, 150°, 390°, 510°\), and only then halve each: \(x = 15°, 75°, 195°, 255°\).
Four solutions instead of two, because the wave completed two cycles in the interval. Halving first and then looking for solutions loses half of them, and this is the single most common error in the topic.
No solutions at all
\(\sin x = 1.4\) has none. Sine is a height on a circle of radius 1, so it never leaves \([-1, 1]\), and neither does cosine. Recognising that immediately saves a fruitless hunt.
Tangent has no such limit — it takes every real value — so \(\tan x = 1000\) does have solutions.
Common mix-ups
- Giving only the calculator’s answer.
- Halving the range before solving when the angle is multiplied.
- Using the sine rule of \(180° - x\) on a cosine equation. Each function has its own symmetry.
- Hunting for solutions to \(\cos x = 2\), which has none.
Check yourself
Solve \(\cos x = 0.5\) for \(0° \le x \le 360°\).
The calculator gives \(60°\). Cosine’s second solution is \(360 - 60 = 300°\). So \(x = 60°\) or \(300°\).
LEVEL 3 · REFLECT
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