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How many solutions does the equation 2*arccos(x) + arcsin(x) = 11*pi/6 have?

1. no solution
2. only one solution
3. two solutions
4. three solutions

Correct answer: no solution

Solution

Since arcsin(x) + arccos(x) = pi/2, we have 2*arccos(x) + arcsin(x) = arccos(x) + pi/2. Setting arccos(x) + pi/2 = 11pi/6 gives arccos(x) = 11pi/6 - pi/2 = 11pi/6 - 3pi/6 = 8pi/6 = 4pi/3. But the range of arccos is [0, pi], and 4pi/3 > pi, so no x satisfies it. Hence no solution.

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