Basic Trigonometric Equations on the Unit Circle
MathematicsTrigonometryAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Trigonometry
Solve sin x = m, cos x = m, or tan x = m using the unit circle: the simulation marks the terminal points that satisfy the equation, compares them with the function graph, and writes out the full solution family x = α + k2π (or π − α + k2π, α + kπ). Change m to see when the equation has no solution.
Lesson: Basic trigonometric equations
What it shows
The unit circle and the function graph are shown side by side to explain why basic trigonometric equations have families of solutions. For sin x = m, the line y = m meets the circle at two points symmetric about the vertical axis: x = α + k2π or x = π − α + k2π. For cos x = m, the points are symmetric about the horizontal axis: x = ±α + k2π. For tan x = m, the two points are π apart, so one family α + kπ is enough. Sine and cosine equations need |m| ≤ 1; tangent equations always have solutions.
How to use
Pick an Equation and start with the quick buttons m = 1/2, m = √2/2 and m = 1, comparing the marked points on the circle with the intersections on the graph. Then slide the Value of m slowly past 1 and −1 and watch the solutions disappear. Switch to tan x = m and repeat. Turn Show function graph off to have students predict the solution family from the circle alone before checking.
Parameters you can change
- Equation sin x = m, cos x = m, tan x = m
- Value of m -2–2
- Show function graph
Questions to explore
- Why does sin x = 1 give only one solution family instead of two?
- Does cos x = 1.5 have a solution, and how does the unit circle show it?
- Why do the solutions of tan x = m repeat every π instead of every 2π?
