Exercises
- What is the radian measure of an angle that measures 37°?
- What is the degree measure of an angle that measures 2 radians? How do you express this degree measure in DMS (degrees/minutes/seconds, °/′/″)?
- What is the degree measure of an angle that measures 31 radians? How do you express this degree measure in DMS (degrees/minutes/seconds, °/′/″)?
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Convert the following degree measurements of an angle to radian measure.
106°98.6°34.5678°17°17′17″76°54′32″
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Without referring to technology,
convert each of following degree measurements of an angle to radian measure,
expressing it as a fraction of \(\pi.\)
Protip: recall that 360° and \(2\pi\) radians
both correspond to a full rotation.
Draw a picture of a circle representing a full rotation,
and think of each degree measurement
as a fraction of a full rotation, and so as a fraction of \(2\pi.\)
30°45°60°90°120°135°150°180°180°210°225°240°270°300°315°330°360°
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Without referring to technology,
for each of the following radian measurements of an angle,
convert it to degree measure.
\(\displaystyle \frac{\pi}{4}\)\(\displaystyle \frac{5\pi}{4}\)\(\displaystyle \frac{11\pi}{6}\)\(\displaystyle \frac{7\pi}{4}\)\(\displaystyle \frac{2\pi}{3}\)\(\displaystyle \frac{\pi}{2}\)\(\displaystyle \frac{3\pi}{4}\)\(\displaystyle \frac{4\pi}{3}\)\(\displaystyle \frac{\pi}{6}\)\(\displaystyle \frac{5\pi}{6}\)\(\displaystyle \frac{\pi}{3}\)
- What is the principal angle coterminal to 1234°? What is the principal angle coterminal to -42°?
- What is the principal angle corresponding to an angle that measures 129 radians? What is the degree measure of this principal angle? How do you express its degree measure in degrees/minutes/seconds?
- Write down three different angles coterminal to 321°, at least one of them having negative measure.
- Consider a sector of a circle of radius 7′ with a central angle of 37°. What is the area of that sector? What is the perimeter of that sector?
- Consider a sector of a circle of radius 1′ that has an outer arc of length 2′. What is the measure of the central angle of this sector?
- Consider a sector of a circle of radius 3′ that itself has an area of 7 ft². What is the measure of the central angle of this sector?
- Explain, as if explaining to a twelve-year-old, the significance of radian measure of an angle. Feel free to compare/contrast it to degree measure.
Problems & Challenges
- The earth is approximately spherical, with a radius of 3960 miles. Classically, a nautical mile was defined to be the distance along the earth’s surface that subtends a central angle of 1′ (one minute, one-sixtieth of a degree). How many international miles (US land miles) long is in a nautical mile?
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Sticking to the convention that our polar axis points eastward (to the right)
and that positive angles correspond to counterclockwise rotation,
every direction on a compass rose corresponds to an angle:
north is 90°, west is 180°, northeast is 45°, and so on.
The directions north (N), south (S), east (E), and west (W)
are called the cardinal directions.
The directions northeast (NE), southeast (SE), northwest (NW), and southwest (SW)
are called the intercardinal directions.
Altogether the cardinal and intercardinal directions
are referred to as the principal winds.
The directions halfway between any two principal winds
are called half-winds —
e.g. east-northeast (ENE) is the half-wind between east and northeast.
The directions halfway between any principal wind
and half-wind are called quarter-winds,
and are named by saying the nearest principal wind
followed by “by” followed by the nearest cardinal direction on its other side —
e.g. northeast-by-east (NEbE) is halfway between northeast and east-northeast.
- What angle does northeast correspond to? What angle does east-northeast correspond to? What angle does east-by-north correspond to?
- What angle does southwest-by-south corresponding to?
