Question

The circular measure of the included angle formed by the hour hand and minute hand of a clock at 3 p.m. will be ___.\[\]

Answer 1

Hint: We recall the functions of the hour hand and minute hand in the watch. We find the angle minute hand will subtend as it passes from one number to another which equivalent to 5 minutes. We then find the measure of the subtended by the minute hand when it passes from 12 to 3 which is also the required angle.

Complete step-by-step solution

We know that in a watch the 12 numbers from 1 to 12 to represent the measurement of time. The hour hand is the shortest hand in the watch which moves from one number to another to represent the passing one hour. The minute hand is the longer hand c which moves from one digit to another to represent the passing of 5 minutes. So when minute hands pass all the 12 numbers it will take a total of $12\times 5=60$ minutes and by that time the hour hand would have moved only to the next number. \[\]
We know that when the minute hand moves to pass all the 12 numbers it subtends a complete angle. The measurement of complete angle in degree is ${{360}^{\circ }}$. We denote the measure of the angle which the minute hand makes passing from one number to another which is the equivalent of 5 minutes as $\theta $. So we have
\[\theta =\dfrac{360}{12}={{30}^{\circ }}\]
We are given the question that the hour hand and minute hand of a clock show 3 PM or AM. So the minute hand will point to 12 and the hour hand will point to 3. We move the minute hand from 12 to 3 while keeping the hour hand unmoved to measure the angle between them. So the minute hand will pass 12 to 1, 1 to 2, and 2 to 3 which is equal to the passing of time in $3\times 5=15$minutes. So the equivalent angle it will be subtended is 2 times the angle it subtends for 5 minutes. So we have the measure of the included angle as
\[3\theta =3\times {{30}^{\circ }}={{90}^{\circ }}\]

Note: We can directly find the required angle as by the formula $\Delta \theta =\left| {{\theta }_{h}}-{{\theta }_{m}} \right|$ where ${{\theta }_{h}}={{0.5}^{\circ }}\left( 60\times H\times M \right)$ and ${{\theta }_{m}}={{6}^{\circ }}\times M$ where ${{\theta }_{h}},{{\theta }_{m}}$ are the angle of hour hand and minute hand measured clockwise from 12 positions, $H$ is the hour and $M$ is the minute passed hour.