How Do You Solve a Trigonometric Equation Graphically in a Finite Interval?

Graph both sides of the equation on the stated finite interval, then use your GDC to find every intersection. The coordinates' $x$-values are the solutions.

The Graphical Method

This is HL-only content from AHL 3.8: solving trigonometric equations graphically. It may appear on any Math AI HL paper, and a GDC is available throughout.

1. Identify the interval and angle unit. On HL papers, assume radians unless a degree symbol is shown.
2. Enter the left-hand side as $Y_1$ and the right-hand side as $Y_2$.
3. Set the graph window so that its horizontal range exactly covers the given interval.
4. Locate every point where the two graphs intersect.
5. Use the GDC intersection function to obtain each $x$-coordinate.
6. Check that every answer belongs to the stated interval and give the required accuracy.

For example, solve

$$\sin x=0.4, \qquad 0\leq x\leq 2\pi.$$

Graph

$$Y_1=\sin x \qquad \text{and} \qquad Y_2=0.4$$

over $0\leq x\leq 2\pi$. The graphs intersect twice, giving

$$x\approx 0.412 \quad \text{or} \quad x\approx 2.73.$$

An equivalent method is to rearrange the equation as $\sin x-0.4=0$, graph $Y_1=\sin x-0.4$, and find all -intercepts.

Why the Window Matters

A poor viewing window can hide solutions or make two nearby intersections appear to be one. Match the horizontal limits to the stated interval, then inspect the graph and use the intersection tool rather than estimating from pixels. If the interval is closed, test both endpoints separately because an endpoint solution may not look like an ordinary crossing. Store precision and round only the final listed values.

Common Misconception

A common mistake is reporting only the principal GDC value, such as $\sin^{-1}(0.4)$. A graphical solution must include every intersection within the finite interval, including endpoints if they satisfy the equation.

Exam Technique

For the IB command term solve, show the equations entered or a labelled sketch before listing the solutions. Keep the GDC in the correct angle mode, use an appropriate viewing window, and report all solutions in mathematical notation, normally exactly or to three significant figures.