Solving inequalities like sin x > a and cos x ≤ a using the unit circle or the graph, on one period or for all real numbers.
The idea: arcs and intervals
To solve \(\cos x > \frac{1}{2}\), you want the points on the unit circle whose \(x\)-coordinate is greater than \(\frac{1}{2}\): everything to the right of the vertical line \(x = \frac{1}{2}\). Those points form an arc, and the arc corresponds to an interval of angles.
Equivalently, you can look at the graph of \(y = \cos x\) and find where it lies above the horizontal line \(y = \frac{1}{2}\).
Tangent
Inequalities that reduce to basic ones
Practice
Quick check
For which \(x\) is \(\sin x \ge -1\)?
Sine is never less than \(-1\).
Quick check
What is the solution of \(\cos x < 0\) on \([0, 2\pi)\)?
Cosine is negative in quadrants II and III.
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