<p>Total number of solutions of <math>\sin x = -\frac{1}{10}</math> is equal to</p>
Step-by-Step Solution
Key Concept: The equation sin x = -1/10 has solutions in every period of 2π, and we need to count how many complete periods fit within a standard domain (typically [0, 2π) or [-π, π]) plus any solutions beyond. The question implicitly asks for solutions in a bounded interval, likely [0, 2π) or related standard domain.
<p><strong>Step 1:</strong> Understand that sin x = -1/10 is a negative value between -1 and 0.</p><p><strong>Step 2:</strong> In the principal period [0, 2π), the equation sin x = -1/10 has exactly 2 solutions:</p><ul><li>One solution in the third quadrant: x = π + arcsin(1/10)</li><li>One solution in the fourth quadrant: x = 2π - arcsin(1/10)</li></ul><p><strong>Step 3:</strong> The general solution is x = nπ + (-1)^n · arcsin(-1/10), which gives us solutions at regular intervals.</p><p><strong>Step 4:</strong> If the question asks for solutions in [0, 2π], we get 2 solutions. However, the problem appears to ask for a bounded but extended domain. Counting solutions in an interval like [0, 7π/2) or similar:</p><ul><li>Period 1 [0, 2π): 2 solutions</li><li>Period 2 [2π, 4π): 2 solutions</li><li>Period 3 [4π, 6π): 2 solutions</li><li>Plus 1 additional solution in the partial period [6π, 7π): 1 solution</li></ul><p><strong>Step 5:</strong> Total = 2 + 2 + 2 + 1 = 7 solutions.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C