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      Question

      Find the number of real solutions of sinx=x

      B 1 Correct Answer Incorrect Answer
      C 2 Correct Answer Incorrect Answer
      D Infinite Correct Answer Incorrect Answer

      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

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