{{ article.displayTitle }}

Dylan always wakes up late for school and misses the bus. He wants to design a clock with an alarm on it. Dylan brings a rough sketch of the design to his Auntie Wilma, an engineer. Seeing the numbers are not accurately placed, she teaches Dylan how to properly distribute the numbers of the clock.

Knowing this, Dylan erases all the numbers on his design and starts over. Auntie Wilma recommends that he draw a coordinate plane on top of the design so that the origin is at the center of the clock.

Solution

Dylan starts by remembering that the initial side of an angle in standard position is on the $x\text{-}$axis. Since the measure of the angle is $0^{\circ },$ the terminal side of the angle lies on top the initial side.

The numeral at $0^{\circ }$ is the one on the right hand side of a clock. By looking at one of Auntie's clocks, Dylan figures out that the numeral is $3.$ He goes ahead and writes it down on his clock design.

According to Auntie Wilma, numerals are $30^{\circ }$ apart from each other. Thus, to find the numeral at $60^{\circ }$ in standard position, Dylan counts how many $30^{\circ }$ intervals are in $60^{\circ }.$

$$\begin{array}{c}60^{\circ }=2\cdot 30^{\circ }\end{array}$$

Auntie Wilma reminds Dylan here that positive angles are measured counterclockwise.

Once more, Dylan checks one of Auntie's clocks and finds that $1$ is the numeral at $60^{\circ }.$ So, he writes it down. So far, his design looks as follows.

Dylan sets out some art supplies to begin painting the clock's housing. He checks another one of Auntie Wilma's clocks as he begins painting. It is $3\text{:}00\text{ PM}.$

Finished and exhausted, Dylan checks the clock again. He is surprised to see that it is now $8\text{:}00\text{ PM}.$ What a long painting session!

Solution

The painting session lasted from $3\text{:}00\text{ PM}$ to $8\text{:}00\text{ PM}.$ This means that it was $5$ hours long.

$$\begin{array}{c}8-3=5\end{array}$$

Since the numerals in the clock are $30^{\circ }$ apart from each other, it is possible to find the displacement of the hour hand of the clock.

$$\begin{array}{c}5\cdot 30^{\circ }=150^{\circ }\end{array}$$

Because the hands of the clock rotate clockwise, the angle measure is negative. This means that the hour hand was displaced by $\text{-}150^{\circ }.$

To find the measure in radians, the relation between radians and degrees will be used.

Relation between Radians and Degrees

$180^{\circ }=\pi \text{ rad}$

This relation can be used to find the displacement of the hour hand in radians.

$180^{\circ }=\pi \text{ rad}$

DivEqn

$\text{LHS}/6=\text{RHS}/6$

$\frac{180^{\circ }}{6}=\frac{\pi }{6}\text{ rad}$

CalcQuot

Calculate quotient

$30^{\circ }=\frac{\pi }{6}\text{ rad}$

MultEqn

$\text{LHS}\cdot (\text{-}5)=\text{RHS}\cdot (\text{-}5)$

$\text{-}5\cdot 30^{\circ }=\text{-}5\cdot \frac{\pi }{6}\text{ rad}$

Multiply

Multiply

$\text{-}150^{\circ }=\text{-}5\cdot \frac{\pi }{6}\text{ rad}$

MoveLeftFacToNum

$a\cdot \frac{b}{c}=\frac{a\cdot b}{c}$

$\text{-}150^{\circ }=\text{-}\frac{5\pi }{6}\text{ rad}$

Consequently, the hour hand rotated $\text{-}\frac{5\pi }{6}$ radians.

Although Dylan has just measured angles counterclockwise, Auntie Wilma suggests that he should be able to measure angles clockwise. Dylan knows that the angle between the $3$ and $1$ numeral measures $60^{\circ },$ his goal is to express that angle in the same direction as the movement of the clock's hands.

Solution

Dylan wants to find an angle that is coterminal to a $60^{\circ }$ angle that is measured in the same direction as the movement of the clock's hands. This means that he is looking to measure it clockwise. That means the measure will be negative in value.

Coterminal angles can be found by adding or subtracting multiples of $360^{\circ }$ to a given angle. Since Dylan is looking for the negative coterminal angle closest to $60^{\circ },$ the coterminal angle can be found by subtracting $360^{\circ }$ from $60^{\circ }.$

$$\begin{array}{c}60^{\circ }-360^{\circ }=\text{-}300^{\circ }\end{array}$$

To find the measure in radians, the relation between radians and degrees will be used.

Relation between Radians and Degrees

$180^{\circ }=\pi \text{ rad}$

This relation can be used to convert $300^{\circ }$ into radians.

$180^{\circ }=\pi \text{ rad}$

DivEqn

$\text{LHS}/3=\text{RHS}/3$

${\frac{180}{3}}^{\circ }=\frac{\pi }{3}\text{ rad}$

CalcQuot

Calculate quotient

$60^{\circ }=\frac{\pi }{3}\text{ rad}$

MultEqn

$\text{LHS}\cdot (\text{-}5)=\text{RHS}\cdot (\text{-}5)$

$\text{-}300^{\circ }=\text{-}\frac{5\pi }{3}\text{ rad}$

Therefore, the coterminal angle measures $\text{-}\frac{5\pi }{3}$ radians.

Dylan is still working on pinpointing the clock numerals. To help himself, he draws a coordinate plane on top of his design, just as Auntie Wilma suggested.

Solution

The terminal side of a quadrantal angle lies on the $x\text{-}$ or $y\text{-}$axis. Therefore, the numerals that Dylan should draw are those that lie on top of the coordinate axes, except for the numeral $3,$ which he drew earlier.

Once more, Dylan looks at one of Auntie's clock and sees that the numerals corresponding to quadrantal angles are $3,$ $6,$ $9,$ and $12.$ However, he already drew $3.$ Hence, he proceeds to draw these three new numerals in his design.

To find the numerals that form reference angles of $30^{\circ },$ Dylan first reviews the definition of reference angle.

Reference Angle

For an angle $\theta$ that is not quadrantal, the acute angle ${\theta }^{\prime }$ formed by the terminal side of $\theta$ and the $x\text{-}$axis is called a reference angle.

Note that this is the angle between the $x\text{-}$axis and the corresponding numeral. This means that it does not have to be an angle in standard position; it can also be either clockwise or counterclockwise.

By looking at the clock on the living room wall of Auntie's house, Dylan finds that the numerals which form a $30^{\circ }$ reference angle are $2,$ $4,$ $8,$ and $10.$ He goes ahead and adds those numerals to the design.

Dylan is ready to finish the clock. He places it, without the housing, in his works place, then sets out a huge sheet of pink graph paper over the already placed numerals. He placed each numeral $1$ foot away from the clock's center.

Dylan wants to place the rest of the numerals, but he has forgotten a protractor to measure the angles! Auntie Wilma points out that his clock can be seen as a unit circle, therefore, the rest of the numerals can be placed by finding their coordinates.

Solution

Dylan begins by finding the angle corresponding to each of the numerals. Going clockwise, the $11$ numeral is $2$ units away from the reference $9$ numeral. Dylan notices that since each numeral is $30^{\circ }$ apart from each other, the angle corresponding to the $11$ numeral can be found.

$$\begin{array}{c}180-2\cdot 30^{\circ }=120^{\circ }\end{array}$$

Dylan thinks that the angle corresponding to the numeral $5$ can be found in a similar way. It is $2$ units away clockwise from the $3$ numeral, which is at $360^{\circ }.$

$$\begin{array}{c}360^{\circ }-2\cdot 30^{\circ }=300^{\circ }\end{array}$$

Auntie Wilma recommends to summarize the information in a table, along with the corresponding reference angle.

Numeral	Angle	Reference Angle
$11$	$120^{\circ }$	$60^{\circ }$
$5$	$300^{\circ }$	$60^{\circ }$

Knowing that the numerals are placed in a unit circle, Dylan will find their coordinates using circular functions.

$$\begin{array}{rl}x& =\cos \theta \\ y& =\sin \theta \end{array}$$

Dylan knows that Auntie Wilma would suggest that this information be added into the table.

Numeral	Angle	Reference Angle	$x\text{-}$coordinate	$y\text{-}$coordinate
$11$	$120^{\circ }$	$60^{\circ }$	$\cos 120^{\circ }=\text{-}\cos 60^{\circ }$	$\sin 120^{\circ }=\sin 60^{\circ }$
$5$	$300^{\circ }$	$60^{\circ }$	$\cos 300^{\circ }=\cos 60^{\circ }$	$\sin 300^{\circ }=\text{-}\sin 60^{\circ }$

Next, using a calculator, Dylans finds the sine and cosine of $60^{\circ }.$ He then rounds each result to two decimal places.

$$\begin{array}{c}\cos 60^{\circ }=0.5\sin 60^{\circ }\approx \text{-}0.87\end{array}$$

Dylan then writes the coordinates of the $11$ and $5$ numerals into the table.

Numeral	$x\text{-}$coordinate	$y\text{-}$coordinate
$11$	$\text{-}0.5$	$\approx 0.87$
$5$	$0.5$	$\approx \text{-}0.87$

Now that Dylan knows the coordinates of the numerals, he proceeds and draws them. Right before drawing the last numeral, $7,$ Auntie Wilma asks him to let her do it, and Dylan happily agrees.

Dylan feels so proud to have made it this far in the clock design. Thanks to viewing his clock as a unit circle, he could draw the last three numerals. Dylan is supremely confident that they are correctly placed. He puts on the finishing touches, removes some of the tics for a more modern look, and sets up this huge and funny looking alarm clock by his bed.