An analog clock of radius 1 foot is positioned on an xy-plane so that its center is at the origin and the 12 o'clock position is on the positive y-axis. How many times per day does the minute hand and hour hand line up perfectly? We will figure this out using a few trigonometric equations. The tip of the minute hand and the tip of the hour hand are points in the plane of the form (Sin (bt), COS (th, where t is a number of minutes after 12 am. When written this way, if = 0 minutes corresponds to 12 am, and the points move clockwise as t increases. Note that this assumes the length of both the minute and hour hands are 1 ft. Though this is not true for analog clocks, it won't make any difference assuming it for this problem. 1. Find the value of b so that (Sin (bt),cos (bt)) gives the position of the minute hand 1." minutes after 12 am. Hint: When bt = 271', the minute hand will have completed one full rotation. What number should we plug in for t to complete one full rotation? 2. Find the value of b so that (Sin (bt),COS (bt)) gives the position of the hour hand t minutes after 12 am. 3. In order for the minute hand and hour hand to be in the same position, the x-values of the points above must be the same. Find all values oft so that the x-values of the above points are equal. Hint: A sum-to-product identity will be helpful here. 4. In order for the minute hand and hour hand to be in the same position, the y-values of the points above must be the same. Find all values of t so that the y-values of the above points are equaL 5. If the hands are to line up, both the x-values and the y-values must be equal. Which t-values appear as solutions to part 3 and solutions to part 4? 6. One full day is 720 minutes. How many t-values from part 5 occur in the interval [0, 720)