Angles and Their Measure

Section 1.2 Angles and Their Measure

One method people use to identify their position is by looking at the latitude. These imaginary lines form circles around the Earth and run parallel to the Equator. The latitude of a place is defined as the angle between a line drawn from the center of the Earth to that point and the equatorial plane. For any point in the Northern Hemisphere, a navigator can measure their latitude by determining the angle that Hōkūpaʻa (also known as Kūmau, Wuli wulifasmughet, Fuesemagut, North Star, or Polaris) makes with the horizon.

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During voyages, knowing the correct angles can make the difference between reaching your destination or missing it. Navigators carefully observe angles on the Hawaiian Star Compass to determine the entry and exit points of celestial bodies in the sky, as well as the direction of wind and current. In this section, we will explore the properties of angles and their measure.

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Definition 1.2.1.

A ray is a part of a line that begins at a point

O

and extends in one direction.

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Definition 1.2.2.

We can create an angle,

θ,

by rotating rays. First, we begin with two rays lying on top of each other and beginning at

O .

We let one ray be fixed and will rotate the second ray about the point

O .

The ray that is fixed is called the initial side and the ray that is rotated is called the terminal side.

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Remark 1.2.3.

Angles are often measured using Greek letters. The commonly used Greek letters include

θ,

ϕ,

α,

β,

and

γ .

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Subsection 1.2.1 Degree

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The measure of an angle is the amount of rotation from the initial side to the terminal side. One unit of measuring angles is the degree. One degree, denoted by

1^{\circ},

\frac{1}{360}

of a complete circular revolution, so one full revolution is

360^{\circ} .

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The Hawaiian Star Compass consists of 32 houses, each spanning

{11.25}^{\circ}

(

\frac{360^{\circ}}{32}

). Assuming due East corresponds to

0^{\circ}

and the center of the House of Hikina points due East, the border between Hikina and Lā Koʻolau will be half the angle of the house,

{5.625}^{\circ}

(

\frac{{11.25}^{\circ}}{2}

). The angles for the other boundaries on the Hawaiian Star Compass are shown in Figure 1.2.4.

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Figure 1.2.4. The Star Compass with the angles indicating the boundaries for each House.

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Although decimals are commonly used to represent fractional parts of a degree, traditionally, degrees were represented in minutes and seconds. One minute or arc minute, denoted as

1^{'},

is equal to

\frac{1}{60}

degrees, and one second or arc second, denoted as

1^{″},

is equal to

\frac{1}{60}

minutes.

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Remark 1.2.5. Conversion Between Degree, Minutes, and Seconds.

\begin{aligned} 1^{\circ} & = 60^{'} & 1^{'} & = {(\frac{1}{60})}^{\circ} & 1^{″} & = {(\frac{1}{3600})}^{\circ} \\ 1^{\circ} & = 3600^{″} & 1^{'} & = 60^{″} & 1^{″} & = {(\frac{1}{60})}^{'} \end{aligned}

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Example 1.2.6. Convert angle from decimal degrees to degrees/minutes/seconds.

In the Star Compass (Figure 1.2.4), the angle between the houses Manu Hoʻolua (northwest) and Noio Hoʻolua (northwest by west) measures

{140.625}^{\circ} .

Represent this angle in degrees, minutes, and seconds.

Solution.

First we will convert

{0.625}^{\circ}

to minutes using the conversion

1^{\circ} = 60^{'},

{0.625}^{\circ} = {0.625}^{\circ} \cdot \frac{60^{'}}{1^{\circ}} = {37.5}^{'}

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Since

1^{'} = 60^{″},

we can convert

{0.5}^{'}

to seconds:

{0.5}^{'} = {0.5}^{'} \cdot \frac{60^{″}}{1^{'}} = 30^{″} .

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{140.625}^{\circ} = 140^{\circ} 37^{'} 30^{″} .

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Example 1.2.7. Convert angle from degrees/minutes/seconds to decimal degrees.

Convert

263^{\circ} 24^{'} 45^{″}

to decimal degrees.

Solution.

We will first convert

24^{'}

and

45^{″}

to degrees.

24^{'} = 24 \cdot 1^{'} = 24 \cdot {(\frac{1}{60})}^{\circ} = {0.4}^{\circ}

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and

45^{″} = 45 \cdot 1^{″} = 45 \cdot {(\frac{1}{3600})}^{\circ} = {0.0125}^{\circ}

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263^{\circ} 24^{'} 45^{″} = 263^{\circ} + 24^{'} + 45^{″} = 263^{\circ} + {0.4}^{\circ} + 0.0125 = {263.4125}^{\circ}

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Definition 1.2.8.

If an angle is drawn on the

x y

-plane, and the vertex is at the origin, and the initial side is on the positive

x

-axis, then that angle is said to be in standard position. If the angle is measured in a counterclockwise rotation, the angle is said to be a positive angle, and if the angle is measured in a clockwise rotation, the angle is said to be a negative angle.

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Definition 1.2.9.

When an angle is in standard position, the terminal side will either lie in a quadrant or it will lie on the

x

-axis or

y

-axis. An angle is called a quadrantal angle if the terminal side lies on

x

-axis or

y

-axis. The two axes divide the plane into four quadrants. In the Cartesian plane, the four quadrants are Quadrant I, II, III, and IV. The corresponding quadrants of Star Compass are Koʻolau (NE), Hoʻolua (NW), Kona (SE), and Malanai (SW).

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Definition 1.2.10.

Coterminal angles are angles in standard position that have the same initial side and the same terminal side. Any angle has infinitely many coterminal angles because each time we add or subtract

360^{\circ}

from it, the resulting angle has the same terminal side.

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Example 1.2.11. Coterminal angles.

90^{\circ}

and

450^{\circ}

are coterminal angles since

450^{\circ} - 360^{\circ} = 90^{\circ} .

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To determine the quadrant in which an angle lies, add or subtract one revolution (

360^{\circ}

) until you obtain a coterminal angle between

0^{\circ}

and

360^{\circ} .

The quadrant where the terminal side lies is the quadrant of the angle. Quadrantal angles do not lie in any quadrant.

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Example 1.2.12. Determine the corresponding house and quadrant in the Star Compass.

Determine the quadrant in which each angle lies and name the corresponding House and quadrant in the Star Compass (Figure 1.2.4).

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(a)

140^{\circ}

Solution.

Since

90^{\circ} < 140^{\circ} < 180^{\circ},

140^{\circ}

lies in Quadrant II, or Manu Hoʻolua.

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(b)

- 770^{\circ}

Solution.

Since

- 770^{\circ} < 0^{\circ},

we first add

3 \times 360

- 770^{\circ}

to obtain an angle

0^{\circ}

and

360^{\circ},

- 770^{\circ} + 3 \times 360^{\circ} = 310^{\circ}

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310^{\circ}

and

- 770^{\circ}

are coterminal. Since

270^{\circ} < 310^{\circ} < 360^{\circ},

310^{\circ}

lies in Quadrant IV, or Manu Malanai.

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(c)

923^{\circ}

Solution.

Since

923^{\circ} > 360^{\circ}

we begin by subtracting

2 \times 360^{\circ}

923^{\circ} - 2 \times 360^{\circ} = 203^{\circ}

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203^{\circ}

and

923^{\circ}

are coterminal. Since

180^{\circ} < 203^{\circ} < 270^{\circ},

923^{\circ}

lies in Quadrant III or ʻĀina Kona.

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Example 1.2.13. Determine the corresponding quadrant given its location in the Star Compass.

What is the corresponding quadrant for Nālani Kona?

Solution.

Locating Nālani Kona in the Star Compass, we see it is in Quadrant III.

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Definition 1.2.14.

A central angle is a positive angle formed at the center of a circle by two radii.

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Remark 1.2.15. Heading and Azimuth.

In navigation, the direction a waʻa is pointed towards is referred to as the heading. Unlike in trigonometry, where it is conventional to define an angle in standard position, i.e.,

0^{\circ}

lies along the positive

x

-axis, in navigation, North corresponds to a heading of

0^{\circ}

and positive angles are measured in a clockwise rotation (see Figure 1.2.16).

Figure 1.2.16. The cardinal directions for headings are as follows: $0^{\circ}$ (or $360^{\circ}$ ) points north, $90^{\circ}$ points east, $180^{\circ}$ points south, and $270^{\circ}$ points west.

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The Star Compass can now be presented in terms of heading angles, as demonstrated in Figure 1.2.17.

Figure 1.2.17. The Star Compass is presented in terms of heading, with the angles indicating the boundaries for each House.

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In astronomy and navigation, the position of a celestial body as it rises or sets on the horizon can be measured using the azimuth, which indicates the direction of celestial objects relative to an observer’s position. Similar to heading, azimuth starts from the north and increases clockwise.

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In navigation, “heading” typically denotes the direction an object like a canoe or wind is pointed, whereas “azimuth” pertains to the angular measurement of celestial bodies on the horizontal plane. Both “heading” and “azimuth” measure angles in degrees, beginning from north and progressing clockwise. Unless specified otherwise to use the heading or azimuth angle (in which case, refer to Figure 1.2.17), this book will use Figure 1.2.4 for the angles of the Star Compass.

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Subsection 1.2.2 Radian

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Another way to measure an angle is with radians, which measure the the arc of a circle that is formed from an angle.

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Definition 1.2.18. Definition of a Radian.

The radian measure of a central angle in a circle is the ratio of the length of the arc on a circle subtended by the angle to the radius. If

r

is the radius of the circle,

θ

is the angle, and

s

is the arc length, then we have the following

θ = \frac{s}{r}

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A radian is abbreviated by rad.

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The measure of a central angle obtained when the length of the arc is also equal to the radius,

r,

is called one radian (1 rad). Similarly, if

θ = 2

rad, then the arc length equals

2 r .

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The circumference of a circle is

C = 2 π r .

This means that the circumference is

2 π \approx 6.28

times the radius. Consequently, if we were to use a piece of string with the length of the radius, we would need six pieces of string plus a fractional piece of the string, as shown in Figure 1.2.19.

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Figure 1.2.19. One rotation of the unit circle is $2 π \approx 6.28$ radians.

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Remark 1.2.20. Relationships Between Degrees and Radians.

If a circle with radius 1 is drawn, it has

360^{\circ},

and the full arc length is the circumference, which is

2 π .

Therefore,the relationship between degrees and radians is:

360^{\circ} = 2 π radian, or 180^{\circ} = π radian

1 radian = \frac{180^{\circ}}{π}

1^{\circ} = \frac{π}{180} radian

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Remark 1.2.21. Converting Between Degrees and Radians.

To convert degree to radians, multiply by $\frac{2 π radians}{360^{\circ}}$ or $\frac{π radian}{180^{\circ}}$
To convert radians to degrees, multiply by $\frac{360^{\circ}}{2 π radians}$ or $\frac{180^{\circ}}{π radian}$

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Example 1.2.22.

Express

45^{\circ}

in radians.

Solution.

45^{\circ} = 45^{\circ} (\frac{2 π radians}{360^{\circ}}) = \frac{π}{4} radians

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Example 1.2.23.

Express

\frac{5 π}{6}

in degrees.

Solution.

\frac{5 π}{6} rad = \frac{5 π}{6} rad (\frac{360^{\circ}}{2 π rad}) = 150^{\circ}

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Using this method, we can obtain Table 1.2.24 of common angles used in trigonometry and the corresponding radian and degree measures.

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Table 1.2.24. Commonly Used Angles in Trigonometry: Degrees and Radians


Radians	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{3 π}{4}$	$\frac{5 π}{6}$	$π$

Degrees	$0^{\circ}$	$30^{\circ}$	$45^{\circ}$	$60^{\circ}$	$90^{\circ}$	$120^{\circ}$	$135^{\circ}$	$150^{\circ}$	$180^{\circ}$

Radians	$π$	$\frac{7 π}{6}$	$\frac{5 π}{4}$	$\frac{4 π}{3}$	$\frac{3 π}{2}$	$\frac{5 π}{3}$	$\frac{7 π}{4}$	$\frac{11 π}{6}$	$2 π$

Degrees	$180^{\circ}$	$210^{\circ}$	$225^{\circ}$	$240^{\circ}$	$270^{\circ}$	$300^{\circ}$	$315^{\circ}$	$330^{\circ}$	$360^{\circ}$

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Subsection 1.2.3 Arc Length

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Recall that the definition of a radian is the ratio of the arc length to the radius of a circle,

θ = \frac{s}{r} .

By rearranging this formula, we can obtain a formula for the arc length of a circle.

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Theorem 1.2.25.

In a circle of radius

r,

the arc length,

s,

subtended by a central angle (in radians),

θ,

s = r θ

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θ

is given in degrees, then

s = 2 π r \cdot (\frac{θ}{360^{\circ}}) .

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Example 1.2.26.

Find the length of an arc of a circle with radius 10 cm subtended by an angle of

2

radians.

Solution.

Using the arc formula we get

s = 10 cm \cdot 2 rad = 20 cm .

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Example 1.2.27.

Kiritimati, also known as Christmas Island, is an atoll in the Republic of Kiribati. Kiritimati’s location west of the International Date Line makes it one of the first places in the world to welcome the New Year, while Hawaiʻi is one of the last places. Although Kiritimati and Molokaʻi share the same longitude at

157^{\circ} 12^{'}

west (meaning Molokaʻi is directly north of Kiritimati), both islands are 24 hours apart. For example, if the time on Oʻahu is 3:00 pm on Thursday, then at that same moment it is 3:00 pm on Friday in Kiritimati. Find the distance between Kiritimati (

1^{\circ} 45^{'}

north latitude) and Molokaʻi (

21^{\circ} 08^{'}

north latitude). Assume the radius of Earth is 3,960 miles and that the central angle between the two islands is the difference in their laititudes.

Solution.

The measure of the central angle between the two islands is

\begin{aligned} θ & = 21^{\circ} 08^{'} - 1^{\circ} 45^{'} \\ = 19^{\circ} 23^{'} \\ = 19^{\circ} + {\frac{23}{60}}^{\circ} \\ \approx {19.3833}^{\circ} \end{aligned}

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To find the distance, we use Theorem 1.2.25 to find the arc length:

s = 2 π r \cdot (\frac{θ}{360^{\circ}}) \approx 2 π \cdot (3, 960 miles) \frac{{19.3833}^{\circ}}{360^{\circ}} \approx 1, 340 miles

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So the distance between Kiritimati and Molokaʻi is approximately 1,340 miles.

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Subsection 1.2.4 Area of a Sector of a Circle

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Definition 1.2.28. Area of a Sector.

The area of the sector of a circle of radius

r

formed by a central angle of

θ

\begin{aligned} A & = \frac{θ}{360^{\circ}} \cdot π \cdot r^{2}, when θ is in degrees \\ A & = \frac{1}{2} r^{2} θ, when θ is in radians \end{aligned}

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Notice the ratio

\frac{θ}{360^{\circ}}

is the proportion of the angle

θ

(in degrees) to one complete circle. Additionally, the circumference of a circle is given by

2 π r .

Therefore, the arc length is simply the proportion of the central angle to the whole circle multiplied by the circumference of the circle.

s = arc length = (proportion of circle) \cdot (circumference) = (\frac{θ}{360^{\circ}}) \cdot (2 π r)

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Similarly, the area of a circle is given by

π r^{2} .

So the area of sector is the proportion of the central angle to the whole circle multiplied by the area of the circle.

A = area of sector = (proportion of circle) \cdot (area of circle) = (\frac{θ}{360^{\circ}}) \cdot (π r^{2})

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Theorem 1.2.29.

Given a circle of radius

r

formed by a central angle of

θ,

then the arc length and area of the sector formed by

θ

can be expressed as the proportion of the angle to the full circle multiplied by the circumference and area of the circle, respectively.

s = (proportion of circle) \cdot (circumference) = (\frac{θ}{360^{\circ}}) \cdot (2 π r)

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and

A = (proportion of circle) \cdot (area of circle) = (\frac{θ}{360^{\circ}}) \cdot (π r^{2})

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Example 1.2.30.

When sailing, Hōkūleʻa cannot make headway by sailing directly into the wind. It can only sail beyond

67^{\circ}

in either direction from the wind (Figure 1.2.31). If Hōkūleʻa sails for 50 miles, what is the area of the sector that cannot be sailed? Round your answer to the nearest square mile.

Figure 1.2.31. Hōkūleʻa cannot sail within $67^{\circ}$ into the direction of the wind.

Solution.

The angle is

θ = 2 \cdot 67^{\circ} = 134^{\circ}

and the radius is

r = 50

miles. So the area is given by

A = \frac{θ}{360^{\circ}} π \cdot r^{2} = \frac{134}{360} π \cdot 50^{2} \approx 2, 923 square miles

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Subsection 1.2.5 Angular Velocity and Linear Speed

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Consider an object moving along a circle as shown below. There are two ways to describe the circular motion of this object: linear speed which measures the distance traveled; and angular speed which measures the rate at which the central angle changes.

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Definition 1.2.32. Linear Speed.

Suppose an object moves along a circle with radius

r

and

θ

(measured in radians) is the angle transversed in time

t .

Let

s

be the distance the object traveled in time

t .

Then the linear speed,

v,

of the object is given by

v = \frac{s}{t}

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Definition 1.2.33. Angular Speed.

Suppose an object moves along a circle. Let

θ

(measured in radians) be the angle transversed by the object in time

t .

The angular speed,

ω,

of the object is given by

ω = \frac{θ}{t}

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Notice that we can rearrange the angular speed to get

θ = ω t .

Since

s

is an arc length, we have

s = r θ,

and thus we can write the linear speed as

v = \frac{s}{t} = \frac{r θ}{t} = \frac{r ω t}{t} = r ω

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Definition 1.2.34. Linear Speed.

Suppose an object moves along a circle with radius

r

and an angular speed

ω

(measured in radians per unit time). Then the linear speed,

v,

of the object is given by

v = r ω

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Example 1.2.35. Une.

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One method a waʻa uses to change direction is with the hoe uli, or the steering paddle. When a sharp turn is needed for maneuvers such as tacking, the steersperson will turn the handle of the hoe uli in a circular motion, as a lever to scoop the paddle in the water and change the heading of a vessel. This move called une (prounced oo-NAY although it is often mispronounced as oo-NEE) literally translates to “lever.”

Figure 1.2.36. A waʻa (canoe) can change directions by rotating the hoe uli (steering sweep) in a process known as une.

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If the steerperson is performing an une at a rate of 25 rotations per minute and the radius of the circular movement is 2 feet, calculate:

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(a)

The angular speed measured in radians per minute.

Solution.

We are given the angular speed is

ω = 25

revolutions per minute. To convert our angular speed to radians per minute, we use the fact that one revolution is

2 π

radians to get

ω = 25 \frac{revolution}{minute} = 25 \frac{rev}{minute} \cdot 2 π \frac{radians}{revolution} = 50 π \frac{radians}{minute}

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Thus the hoe uli is moving at an angular speed of

50 π

radians per second.

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(b)

The linear speed of the hoe uli in miles per hour (round your answer to two decimal places).

Solution.

Since the radius is

r = 2

ft and the angular speed is

50 π

radians per minute, we can use Definition 1.2.34 to calculate the linear speed

v = r ω = 2 ft \cdot 50 π \frac{rad}{min} \cdot \frac{mile}{5280 ft} \cdot \frac{60 min}{hr} \approx 3.57 \frac{miles}{hour}

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Thus the steersperson is moving the hoe uli at a linear speed of 3.57 mph.

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Exercises 1.2.6 Exercises

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Exercise Group.

Given an angle,

θ,

identify the house and quadrant on the Hawaiian Star Compass.

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1.

θ = 336^{\circ}

Section 1.2 Angles and Their Measure

Definition 1.2.1.

Definition 1.2.2.

Remark 1.2.3.

Subsection 1.2.1 Degree

Remark 1.2.5. Conversion Between Degree, Minutes, and Seconds.

Example 1.2.6. Convert angle from decimal degrees to degrees/minutes/seconds.

Solution.

Example 1.2.7. Convert angle from degrees/minutes/seconds to decimal degrees.

Solution.

Definition 1.2.8.

Definition 1.2.9.

Definition 1.2.10.

Example 1.2.11. Coterminal angles.

Example 1.2.12. Determine the corresponding house and quadrant in the Star Compass.

(a)

Solution.

(b)

Solution.

(c)

Solution.

Example 1.2.13. Determine the corresponding quadrant given its location in the Star Compass.

Solution.

Definition 1.2.14.

Remark 1.2.15. Heading and Azimuth.

Subsection 1.2.2 Radian

Definition 1.2.18. Definition of a Radian.

Remark 1.2.20. Relationships Between Degrees and Radians.

Remark 1.2.21. Converting Between Degrees and Radians.

Example 1.2.22.

Solution.

Example 1.2.23.

Solution.

Subsection 1.2.3 Arc Length

Theorem 1.2.25.

Example 1.2.26.

Solution.

Example 1.2.27.

Solution.

Subsection 1.2.4 Area of a Sector of a Circle

Definition 1.2.28. Area of a Sector.

Theorem 1.2.29.

Example 1.2.30.

Solution.

Subsection 1.2.5 Angular Velocity and Linear Speed

Definition 1.2.32. Linear Speed.

Definition 1.2.33. Angular Speed.

Definition 1.2.34. Linear Speed.

Example 1.2.35. Une.

(a)

Solution.

(b)

Solution.

Exercises 1.2.6 Exercises

Exercise Group.

1.

2.

3.

4.

5.

6.

7.

8.

9.

Exercise Group.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

Exercise Group.

20.

21.

22.

23.