Explore the solutions of trigonometric equations sin(x) = a and cos(x) = b using interactive visualizations. Adjust the parameter values and see how the solutions change on both the unit circle and the function graph.
sin(x) = 0.5
Parameter value: 0.5
-11
Unit Circle
Function Graph (0 to 2π)
Solutions in [0, 2π)
Interactive Tutorial
Use the slider to change the value of the parameter (a for sin(x) = a, b for cos(x) = b).
Click on the equation buttons to switch between sin(x) = a and cos(x) = b.
Use the preset buttons to quickly set special values like 1, -1, 1/2, -1/2.
Observe how the solutions (intersection points) change on both the unit circle and the function graph.
For sin(x) = a, note there are:
No solutions when |a| > 1
One solution when a = 1 or a = -1 (at x = π/2 and x = 3π/2 respectively)
Two solutions when -1 < a < 1 (symmetric about π/2 for sin)
For cos(x) = b, note there are:
No solutions when |b| > 1
One solution when b = 1 or b = -1 (at x = 0 and x = π respectively)
Two solutions when -1 < b < 1 (symmetric about 0 for cos)
Try to relate the solutions on the unit circle (intersections with horizontal/vertical lines) to the solutions on the function graph (intersections with horizontal lines).