Counting Solutions Graphically

The question isn't always "solve sin x = k." Sometimes it is "for how many values of x in [0, 4π] does sin x = k hold?" That is a counting problem, not a solving problem. The tool is a graph, not algebra.

The technique

Draw y = sin x and y = k on the same axes. Count the points where they intersect. Each intersection is one solution.

Drag the line below and watch the count change.

What to notice

• For k = 1 or k = −1: exactly one solution per period (at the peak or trough)
• For −1 < k < 1, k ≠ 0: exactly two solutions per period
• For k = 0: exactly one solution per period (at the zero crossings — but two zero crossings per period, so careful with endpoints)
• For |k| > 1: no solutions

Applying to a restricted domain

Problem: How many solutions does sin x = 0.5 have in [0, 3π]?

Draw y = sin x on [0, 3π]. This is 1.5 periods. The line y = 0.5 cuts the first arch twice (x = π/6 and x = 5π/6), cuts the second arch not at all (second arch is negative), cuts the third half-arch once (x = π/6 + 2π = 13π/6). Total: 3 solutions.

The sketch takes 10 seconds. Algebra alone takes 2 minutes and is prone to missing cases.

Beyond horizontal lines — nonlinear equations

The same idea extends to equations like:

$\sin x = \frac{x}{\pi}$

Draw y = sin x and y = x/π on the same axes. The line through the origin with slope 1/π cuts the sine curve at x = 0 and at two other points (one in each of the first positive and first negative arches, since the slope 1/π ≈ 0.318 is less than the maximum slope of sin x at the origin, which is 1). Three solutions total.

Change to sin x = x/(2π): the slope is now 1/(2π) ≈ 0.159, which cuts even more arches. The line is shallower so it stays inside the ±1 band for longer, cutting more arches before escaping. Five solutions.

The pattern: as the slope of the line decreases, it cuts more arches. As slope increases past 1 (the maximum slope of sin x), the line only intersects at the origin.

Class of problems this solves

Any equation of the form:

• f(x) = k (count intersections with horizontal line)
• f(x) = g(x) where one is a trig function and the other is linear (count intersections of two curves)
• Number of real roots of a trig equation in a given interval

JEE Advanced 2016: How many solutions does cos(sin x) = sin(cos x) have in [0, π]? Draw both sides. Compare ranges. This is a graphical-reasoning problem, not an algebraic one.

---

Next: Composite trig — what happens when sin and cos are composed, and why the range matters more than the shape.