Solving Trig Equations (AA HL)
A trig equation like \(\sin x=\tfrac12\) doesn't have just one answer - because sine and cosine repeat every \(2\pi\), there's usually a whole family of solutions, and the question restricts you to a finite interval so the list stays manageable. This page is about finding every solution in that interval, not just the first one a calculator gives you. It's part of the broader Identities & Equations topic.
32 questions on this sub-topic.
Two ways to solve
Covered under IB syllabus reference SL3.8: solving trigonometric equations in a finite interval, both graphically and analytically. There's no single formula here - it's a technique, and either method below is acceptable unless the question specifies one.
Analytically
Isolate the trig ratio, find one solution using exact-value recall or \(\arcsin/\arccos/\arctan\), then use the symmetry of the sine/cosine/tangent graph to list every other solution inside the interval.
Graphically
Plot \(y=\) each side of the equation on your GDC over the given interval and read off the \(x\)-coordinates of every intersection point. Best when the equation doesn't reduce to a standard angle.
Need the full syllabus wording and formula-booklet reference table? See Identities & Equations.
Worked examples
Solve \(\sin x=\tfrac12\) for \(0\le x\le2\pi.\)
(a)(i) Give the solution with \(x<1.571\).
(a)(ii) Give the solution with \(x>1.571.\)
Worked solution
\(\sin x = \tfrac12 \Rightarrow x = \dfrac{\pi}{6}\) M1 A1 or \(x = \dfrac{5\pi}{6}.\) A1
Using \(3\sin x+4\cos x=5\sin(x+0.927)\), solve \(3\sin x+4\cos x=2\) for \(0\le x<2\pi.\)
Worked solution
\(5\sin(x + 0.927) = 2 \Rightarrow \sin(x + 0.927)\) M1 \(= 0.4.\) A1
\(x + 0.927 = 0.4115\) or \(\pi - 0.4115\) M1 \(= 2.730.\) A1
\(x \approx 1.80\) A1 and \(x \approx 5.77.\) A1
On the GDC: graph \(y = 3\sin x + 4\cos x\) and \(y = 2\) on \([0, 2\pi)\) and read off the intersections, or use the equation solver.
Common mistakes
- Dropping solutions when solving in an interval. A calculator or exact-value recall gives you one angle - always check the symmetric case(s) from the same equation and confirm every valid angle in the interval is listed.
- Forgetting to switch calculator mode. IB exams mix degrees and radians across papers - working in the wrong mode gives an answer that looks plausible but is completely wrong.
- Stopping after finding \(x+c\) instead of \(x\). When the equation is phrased as \(f(x+c)=k\), the values you find first are for the bracket, not for \(x\) - subtract \(c\) from each before writing your final answers.
Ready to practise properly?
33 trig-equation questions, marked instantly like the real exam.
Quick answers
How many solutions does a trig equation have in a given interval?
Usually more than one. Sine and cosine repeat every \(2\pi\) (tangent every \(\pi\)), so once you find one solution from a calculator or exact-value recall, you must generate every other angle in the stated interval using the symmetry of the graph before you're done.
Should I solve a trig equation graphically or analytically?
Either is valid unless the question specifies. Analytically is faster for exact-value angles; graphically (plotting both sides and finding intersections on a GDC) is safer for equations with awkward decimals or that don't reduce to a standard form. See the parent topic's GDC guidance for calculator steps.