Trigonometry question

Question

A clock face is in the form of a coordinate system with the origin at the center of the face. The minute hand passes through the point (1, √3). The hour hand is somewhere between 4 and 5. What time is it?

Answer 1

In a trigonometric circle, the minute hand passes through the point ( 1, √3), indicating that with both points positive the minute hand occurs in the first quadrant, the angle of rotation θ in degrees between the positive horizontal axis and the minute hand where the hand represents the hypotenuse of the right angle triangle is:

θ = tan^-1( √3/1) = 60° rotating counterclockwise from the positive x axis. if the clock between 12 and 3 is in the first quadrant of the circle, it can be divided into 3 equal angles of 30° with the 0° pointing at 3 o'clock, 30° pointing at 2 o'clock, and 60° pointing at 1 o'clock and knowing that there are 15 minutes between 12 and 3 in terms of minutes, then the 60° represent 5 minutes past 12, with the hr hand between 4 and 5 , then the time is past 4 and not yet 5. The time is therefore 4:05 or five minutes past 4

Also note that the right angle triangle formed by the minute hand is a 60,30 triangle with the hypotenuse = c and with c² = a² + b² = ( √3)² + 1² = 4, and c = 2 is also the radius of the circle