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Chapter 4: Problem 19

What is the angle between the hour hand and the minute hand on a clock at \(1: 23 ?\)

Short Answer

Expert verified
The angle between the hour hand and the minute hand on a clock at 1:23 is approximately \(67.00^{\circ}\).

Step by step solution

01

Calculate the angle covered by the hour hand

To calculate the angle covered by the hour hand, first find the fraction of the hour that has passed, which is given by \(\frac{Hour + (\frac{Minutes}{60})}{12}\). Then multiply this fraction by \(360^{\circ}\). In this case, the hour is 1 and the minutes are 23, so the fraction of the hour passed will be calculated as follows: \[ \frac{1 + \frac{23}{60}}{12} = \frac{1 + \frac{23}{60}}{12} \] Now, multiply this fraction by \(360^{\circ}\) to get the angle covered by the hour hand: \[ Hour\, Hand\, Angle \, (HHA) = \frac{1 + \frac{23}{60}}{12} \times 360^{\circ} \]
02

Calculate the angle covered by the minute hand

To calculate the angle covered by the minute hand, find the proportion of the hour that has passed in minutes and divide it by 60, then multiply by \(360^{\circ}\). With 23 minutes passed: \[ Minute \, Hand \, Angle \, (MHA) = \frac{23}{60} \times 360^{\circ} \]
03

Calculate the angle between the hour and minute hands

To find the angle between the hour and minute hands, calculate the absolute difference between the angles covered by each hand. If the difference is greater than \(180^{\circ}\), subtract it from \(360^{\circ}\): \[ Angle \, Between \, Hands = \begin{cases} |HHA - MHA| & \text{if} \ |HHA - MHA| \le 180^{\circ} \\ 360^{\circ} - |HHA - MHA| & \text{if} \ |HHA - MHA| > 180^{\circ} \end{cases} \] Now, plug in the values you found in Step 1 and Step 2 for the hour and minute hand angles, and find the angle between the hands at 1:23.
04

Final answer

After calculating, we get: \[ Hour\, Hand\, Angle = \frac{1 + \frac{23}{60}}{12} \times 360 ^{\circ} \approx 71.00^{\circ} \] \[ Minute\, Hand\, Angle = \frac{23}{60} \times 360 ^{\circ} \approx 138.00^{\circ} \] So the angle between the hands at 1:23 is: \[ Angle\, Between\, Hands = |71.00^{\circ} - 138.00^{\circ}| = 67.00^{\circ} \] Thus, the angle between the hour hand and the minute hand on a clock at 1:23 is approximately \(67.00^{\circ}\).

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Most popular questions from this chapter

In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sin \frac{25 \pi}{12}\)

Show that $$\sin \left(t+\frac{\pi}{2}\right)=\cos t$$ for every number \(t\).

Given that $$\cos 15^{\circ}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ Find exact expressions for the indicated quantities. [These values for \(\cos 15^{\circ}\) and \(\sin 22.5^{\circ}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sec 15^{\circ}\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\sin u\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with $$\tan u=-2 \text { and } \tan v=-3$$ Find exact expressions for the indicated quantities. \(\cos (u+4 \pi)\)
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