Find the number of real solutions of sinx=x

Solution

We are given the equation: sin(x) = x We are to find how many real solutions this equation has. We are comparing two functions:

• f₁(x) = sin(x) — bounded between [–1, 1], periodic, continuous
• f₂(x) = x — an unbounded straight line

We are solving for x where sin(x) = x, i.e., the points of intersection of the graphs of y = sin(x) and y = x. Consider the domain Since sin(x) ∈ [–1, 1], the only possible real solutions must lie within: x ∈ [–1, 1] Outside this interval, sin(x) stays bounded between –1 and 1, but x grows  unbounded ⇒ no solution beyond x ∈ [–1, 1]. Graphical Insight Plotting y = sin(x) and y = x:

• At x = 0: sin(0) = 0 ⇒ x = 0 is a solution
• For x ≠ 0:
• For x in (0, 1]: sin(x) < x
• For x in [–1, 0): sin(x) > x

So, sin(x) = x intersects the line only once, at x = 0