Solving Basic Trig Equations (AA SL)
A basic trig equation gives you one ratio - \(\sin x\), \(\cos x\) or \(\tan x\) equals a number - and asks for every angle in a stated domain that produces it. The skill is entirely about turning that ratio back into an angle without losing any of the valid solutions. This page walks through the reference-angle method with worked examples and the mistakes that cost marks. It's part of the broader Trig Equations & Graphs topic.
20 questions on this sub-topic.
The two-step method
Covered under IB syllabus reference SL3.8. There's no formula-booklet formula for this - it's a fixed two-step method you apply to any \(\sin x = k\), \(\cos x = k\) or \(\tan x = k\) equation.
Step 1: principal value
\(x = \sin^{-1}(k)\), \(\cos^{-1}(k)\) or \(\tan^{-1}(k)\)
Your GDC (or known exact value) gives one solution. This is the value the calculator always returns first, whatever the actual domain is.
Step 2: other solutions in range
Use graph symmetry to find the rest
Sine and cosine repeat every \(2\pi\) and are symmetric about \(\tfrac{\pi}{2}\) and \(\pi\) respectively; tangent repeats every \(\pi\). Add or reflect the principal value using these symmetries until you've covered the whole domain.
Need the full syllabus wording and worked equations-and-graphs overview? See Trig Equations & Graphs.
Worked examples
Solve \(\sin x = \dfrac{1}{2}\) for \(0^{\circ} \le x \le 90^{\circ}\).
Worked solution
Solve in the first quadrant. Take the inverse sine of both sides: \(x=\sin^{-1}(\tfrac12)\) M1 \(=30^\circ\); this is the only solution in \([0^\circ,90^\circ]\). A1
Solve \(\sin x=\dfrac{\sqrt3}{2}\) for \(0\le x\le 2\pi.\)
(a)(i) Give the solution with \(x<1.571\).
(a)(ii) Give the solution with \(x>1.571.\)
Worked solution
\(\sin x=\tfrac{\sqrt3}{2}\Rightarrow\) reference angle \(\tfrac{\pi}{3}\). M1 Sine is positive in Q1 and Q2: R1 \(x=\tfrac{\pi}{3}\) or \(x=\pi-\tfrac{\pi}{3}=\tfrac{2\pi}{3}.\) A1A1
Common mistakes
- Stopping after the principal value. \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\) on your calculator only ever hand back one angle - the domain almost always contains a second (or third, or fourth) solution you have to find yourself using symmetry.
- Guessing the sign in the wrong quadrant. Sine is positive in quadrants 1 and 2, cosine in 1 and 4, tangent in 1 and 3 - mixing these up gives a solution that satisfies the wrong equation entirely.
- Leaving the calculator in the wrong angle mode. A domain given in radians solved in degree mode (or vice versa) produces numbers that look plausible but are completely wrong - check the mode before you start.
Ready to practise properly?
20 basic-trig-equation questions, marked instantly like the real exam.
Quick answers
How do you solve a basic trig equation like sin x = 0.5?
Take the inverse trig function to get the principal value, then use the symmetry of the sine, cosine or tangent graph to find any other solutions that lie inside the given domain.
How many solutions does a basic trig equation have?
It depends on the domain given. Over one full period, \(\sin x = k\) and \(\cos x = k\) usually give two solutions (unless \(k\) is at a maximum or minimum), while \(\tan x = k\) gives one. Wider domains give proportionally more.