Michael keeps it real

Algebra Level 4

For a uniformly random real number \(k\) in the interval \([-10, 0],\) the probability that \(\sin^4 \theta - \sin^2 \theta - k = 0\) has at least one real solution \(\theta\) can be expressed as \(\dfrac{a}{b}\) for positive coprime integers \(a\) and \(b.\) What is the value of \(a+b\)?

This problem is posed by Michael T.

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