The document discusses using trigonometric identities and proportions to solve for the time given the positions of the hour and minute hands on a clock. It first uses the sine sum identity to solve for the hour hand position in terms of hours. It then sets up a proportion to solve for the specific hour. This process is repeated for the minute hand using the cosine sum identity and a proportion to solve for minutes. Both sections solve for the time variables in terms of the hand positions.

1. Annotations 1 The Clock..

2. Slide 1 This equation represents the hour hand of the clock This equation is the Sine Sum Identity We substitute the values into the Sine Sum Identity Since and on the unit circle, we substitute them in, then we simplify to find the values of Since the given interval is the only possible solution is

3. Slide 2 Here we set up an Equation based on Proportions by making the position of the hour hand: equal to the hour of the time: We place X over 12 because there are 12 hours in the after noon, from 12:00 p.m. to 12:00 a.m. stated in the question Next we cross multiply Next, we simplify to find the hour of the time

4. Slide 3 This equation represents the minute hand of the clock This equation is the Cosine Sum Identity We substitute the values into the Cosine Sum Identity Since and on the unit circle, we substitute them in, then we simplify to find the values of Since the given interval is the only possible solution is

5. Slide 4 Here we set up an Equation based on Proportions by making the position of the minute hand: equal to the minutes of the time: We place X over 60 because there are 60 minutes in one hour Next we cross multiply Next we simplify to find the minutes of the time