give graph "cos(θ)<1/2 when θ is between these two angles (but not equal to them)"

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Let's solve the inequality cosine theta is less than one half. We need to find the angles where the cosine function falls below the horizontal line y equals one half. First, we'll graph the cosine function and the line y equals one half to visualize the solution. Now we find where the cosine function intersects the line y equals one half. Setting cosine theta equal to one half, we get theta equals pi over 3 and theta equals 5 pi over 3. These are our critical points where the cosine function equals exactly one half. Now we identify where cosine theta is less than one half. Looking at the graph, the cosine curve falls below the horizontal line between pi over 3 and 5 pi over 3. This shaded region shows where the inequality is satisfied. Since we have a strict inequality, the endpoints are not included in the solution. Let's verify our solution using the unit circle. On the unit circle, the x-coordinate represents the cosine value. The vertical line at x equals one half shows where cosine equals one half. The two intersection points occur at pi over 3 and 5 pi over 3. The highlighted arc between these points represents all angles where cosine is less than one half. To summarize our solution: The inequality cosine theta less than one half is satisfied when theta is between pi over 3 and 5 pi over 3. This is written as pi over 3 less than theta less than 5 pi over 3, where the endpoints are excluded because we have a strict inequality. This interval represents the complete solution to our problem.