Trigonometric Equations

Solve equations involving trigonometric terms. Visualize how horizontal lines intersect periodic waveforms, yielding infinite sets of general solutions.

Interactive Sandbox

waveform intersection

Select an equation preset. Use the target slider to vary constant $c$. Watch the intersections (solutions in $[0, 2\pi]$) update in real-time. If the line lies outside the curve, notice the empty intersection set.

Graph: y = sin(x) vs y = c

Active Equation

sin(x) = 0.50
Target Value (c) 0.50
Pulsing red dots indicate valid coordinates (x, y) where the trigonometric function equals c.
Algebraic Solver

Algebraic Solver

Trace how we compute primary, quadrant-based, and general solutions for the active wave-constant intersections.

Algebraic Expansion Steps


Roots in range [0, 2π]


Numeric Solutions

No Solutions in [0, 2π]

Evaluation

Conceptual Quiz

Test your understanding of solutions frequency, general periodic formula structures, and quadratic trig factorizations.

Question 1 of 5 Score: 0/0

How many solutions does the equation sin(x) = 0.5 have in the interval [0, 2π]?

Correct Answer!

Explanation text...

Solving Cheat Sheet


  • sin(x) = c: x = sin⁻¹(c)  |  x = π - sin⁻¹(c)
  • cos(x) = c: x = cos⁻¹(c)  |  x = 2π - cos⁻¹(c)
  • General Solution (sine): θ = θ₀ + 2kπ
  • Factor: 2sin²x - sinx - 1 = (2sinx + 1)(sinx - 1)