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How many solutions does the equation 16(sin⁵(x) + cos⁵(x)) = 11(sin(x) + cos(x)) have in the interval [0, 2*pi]?

1. 6
2. 7
3. 8
4. 9

Correct answer: 6

Solution

Setting t = sin(x)+cos(x), the equation splits into t = 0 (giving 2 solutions) and t = +/-sqrt(6)/2 (giving 2+2 solutions), for a total of 6 in [0, 2*pi].

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