Trig Equations & Graphs (AA SL)
Sine and cosine graphs describe anything that repeats - a wheel turning, a tide rising and falling, a voltage oscillating. This topic covers reading the amplitude, period and centre line off a trigonometric graph, sketching transformations of the form \(f(x)=a\sin(b(x+c))+d\), and solving trigonometric equations over a given interval both graphically and algebraically.
What the syllabus says
This topic maps onto two points in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL3.7 | The circular functions \(\sin x\), \(\cos x\) and \(\tan x\): amplitude, their periodic nature, and their graphs. Composite functions of the form \(f(x)=a\sin(b(x+c))+d\). Transformations of trigonometric graphs. Real-life contexts such as the height of a tide or the motion of a Ferris wheel. |
| SL3.8 | Solving trigonometric equations in a finite interval, both graphically and analytically, e.g. \(2\sin x = 1\), \(0\le x\le 2\pi\). Equations leading to a quadratic in \(\sin x\), \(\cos x\) or \(\tan x\). The general solution of trigonometric equations is not required. |
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is amplitude?
Amplitude is how far a trigonometric graph swings above (and below) its centre line - it's the coefficient \(a\) in \(f(x)=a\sin(b(x+c))+d\). A larger amplitude means a taller wave; amplitude is always taken as positive.
e.g. \(f(x)=4\sin(2x)-1\) has amplitude \(4\).
What is the period of a trig function?
The period is how far along the x-axis you travel before the graph repeats exactly. For \(f(x)=a\sin(b(x+c))+d\) the period is \(\dfrac{2\pi}{b}\) (in radians) - a larger \(b\) compresses the graph and shortens the period.
e.g. \(f(x)=\sin(3x)\) has period \(\dfrac{2\pi}{3}\approx2.094\).
What is the principal axis?
The principal axis (or centre line) is the horizontal line the graph oscillates about - it's \(y=d\) in \(f(x)=a\sin(b(x+c))+d\). The maximum value is \(d+|a|\) and the minimum is \(d-|a|\).
e.g. \(f(x)=5\sin x + 2\) oscillates about the principal axis \(y=2\).
What does it mean to solve a trig equation?
Solving a trig equation means finding every value of \(x\) within a given interval that makes the equation true - not just one. Because sine and cosine repeat, most trig equations have several solutions in a typical interval like \(0\le x\le 2\pi\).
e.g. \(2\sin x = 1\) for \(0\le x\le 2\pi\) gives \(x=\dfrac{\pi}{6}\) or \(x=\dfrac{5\pi}{6}\).
What is a radian?
A radian is an alternative unit for measuring angles, defined so that \(2\pi\) radians makes a full turn (instead of \(360°\)). AA works in radians throughout this topic, so \(180°=\pi\) radians.
e.g. \(60°=\dfrac{\pi}{3}\) radians, since \(180\div60=3\).
Key formulas
None of these are separate formula-booklet entries - they all fall straight out of reading the composite form \(a\sin(b(x+c))+d\), which is what makes this topic more about pattern recognition than memorisation.
Formula reference
None of the values below are listed as their own formula in the booklet - they're read directly off the function \(f(x)=a\sin(b(x+c))+d\), which is prior knowledge you're expected to apply.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x)=a\sin(b(x+c))+d\) | General sinusoidal function | Not in booklet - definitional |
| Amplitude \(=|a|\) | Height of the wave above/below the centre line | Not in booklet |
| Period \(=\dfrac{2\pi}{|b|}\) | Length of one full cycle | Not in booklet |
| Principal axis \(y=d\) | Centre line of the graph | Not in booklet |
| radians \(=\) degrees \(\times\dfrac{\pi}{180}\) | Converting between angle units | Not in booklet - prior knowledge |
Degrees vs radians
AA works almost exclusively in radians for this topic, but it's worth keeping the two units straight.
| Feature | Degrees | Radians |
|---|---|---|
| Full turn | \(360°\) | \(2\pi\) |
| Half turn | \(180°\) | \(\pi\) |
| Quarter turn | \(90°\) | \(\dfrac{\pi}{2}\) |
| Converting | multiply by \(\dfrac{180}{\pi}\) | multiply by \(\dfrac{\pi}{180}\) |
| Used in AA topic 3 | Rarely | Almost always |
Reading a graph
Every feature of \(f(x)=a\sin(b(x+c))+d\) can be read straight off the equation - no graphing needed.
Amplitude
\[\text{amplitude}=|a|\]
Read directly from the coefficient in front of sin or cos.
Not in the formula booklet - prior knowledgePeriod
\[\text{period}=\dfrac{2\pi}{b}\]
A bigger \(b\) squeezes the graph horizontally, shortening the period.
Not in the formula booklet - prior knowledgePrincipal axis
\[y=d\]
Max \(=d+|a|\), min \(=d-|a|\) - both follow once \(d\) and \(a\) are known.
Not in the formula booklet - prior knowledgeSolving trig equations
The same equation can be tackled graphically or algebraically - both are accepted IB methods.
Graphical method
Graph both sides of the equation as separate functions and find every intersection in the given interval. Fast, and hard to miss a solution if the window is wide enough.
Algebraic method
Isolate \(\sin x\), \(\cos x\) or \(\tan x\), find the principal value with inverse trig, then use the symmetry of the graph to generate every other solution in the interval.
Multiple solutions
Sine and cosine are periodic, so an equation like \(2\sin x=1\) on \(0\le x\le2\pi\) typically has two (or more) solutions - always check the whole interval, not just the first root you find.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Consider \(f(x)=4\cos(3x)-1\).
(a) State the amplitude.
(b) State the period.
(c)(i) State the maximum value.
(c)(ii) State the minimum value.
Worked solution
(a) Amplitude \(=|4|=4.\) A1
(b) Period \(=\dfrac{2\pi}{3}.\) A1
(c)(i) Centre line \(y=-1\): max \(=-1+4=3.\) A1
(c)(ii) Min \(=-1-4=-5.\) A1
Solve \(\cos 2x=\tfrac{\sqrt3}{2}\) for \(0\le x\le 2\pi\).
(a) State the smallest solution.
(b) State the second solution.
(c) State the third solution.
(d) State the largest solution.
Worked solution
(a) Let \(u=2x,\) \(0\le u\le4\pi.\) \(\cos u=\tfrac{\sqrt3}{2}\Rightarrow u=\tfrac{\pi}{6},\tfrac{11\pi}{6},\tfrac{13\pi}{6},\tfrac{23\pi}{6}.\) M1
\(u=\tfrac{\pi}{6},\tfrac{11\pi}{6},\tfrac{13\pi}{6},\tfrac{23\pi}{6}.\) A1
So \(x=\tfrac{\pi}{12},\tfrac{11\pi}{12},\tfrac{13\pi}{12},\tfrac{23\pi}{12}.\) A1
(b) \(x=\tfrac{11\pi}{12}.\) A1
(c) \(x=\tfrac{13\pi}{12}.\) A1
(d) \(x=\tfrac{23\pi}{12}.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Leaving the GDC in the wrong angle mode. AA works in radians throughout this topic - a degree-mode graph or solve gives a completely different (and wrong) answer.
- Stopping after one solution. \(2\sin x=1\) on \(0\le x\le2\pi\) has two solutions, \(\tfrac{\pi}{6}\) and \(\tfrac{5\pi}{6}\) - always check the whole interval before you finish.
- Confusing amplitude with the maximum value. Amplitude is \(|a|\), the height above the centre line - the actual maximum is \(d+|a|\), not \(|a|\) alone.
- Misreading the horizontal shift. In \(a\sin(b(x+c))+d\) the shift is \(-c\), and \(b\) affects both the stretch and the shift together - don't just read \(c\) off as the shift without checking the brackets.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The single most common source of lost marks in trig - your answer is only correct if the angle mode matches the question. AA usually uses radians; AI usually uses degrees.
- Check the question: are angles in degrees (°) or radians (π, rad)?
- Press MODE, highlight RADIAN or DEGREE on the angle row and press ENTER, then 2nd → MODE to quit.TI-84
- Press ctrl → menu → Settings (or doc Settings) → Angle, and choose Degree or Radian.Nspire
- Press SHIFT → MENU (SET UP) → Angle, then choose Deg or Rad and EXIT.Casio
- Re-enter the calculation after switching - the mode only affects new work.
Tip: If a sin/cos/tan answer looks wildly wrong, the angle mode is almost always the cause.
Faster and safer than algebra for messy equations - and essential when many equations can't be solved by hand. The trick is getting all solutions, not just one.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(x) = 0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.Casio
- Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
Tip: For equations like \(\sin x = 0.5\) over an interval, use the graph plus intersection method rather than the equation solver - it's faster and less likely to miss roots.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Trig equations & graphs questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between amplitude and period?
Amplitude measures how far the graph swings above and below its centre line - it's the coefficient in front of sin or cos. Period measures how long, along the x-axis, one full wave takes before it repeats. They describe different directions: amplitude is vertical, period is horizontal.
How many solutions should I find when solving a trig equation?
As many as fit inside the given interval - sin and cos repeat, so most equations have more than one solution. Sketch the graph first (or use the GDC) to see how many crossings there are before you stop looking.
Do I use degrees or radians in AA?
Radians, almost always. AA SL and HL work in radians throughout topic 3, so set your GDC's angle mode to radian before graphing or solving - a degree-mode answer to a radian-mode question will be completely wrong.
Can I use my GDC to solve trig equations?
Yes - graphing both sides and finding intersections, or using the equation solver, is a fully valid IB method on Paper 2. You still need to know how to set up the equation and interpret the interval by hand. See the GDC guide for model-specific instructions.
Sub-topics
Trig Equations & Graphs broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA SL syllabus unit, in case you want to keep going.