What is 3 8 of a Full Rotation

What is 3 8 of a Full Rotation.

Angles — The Revolution

Which is the all-time unit of measure for angles — revolutions, degrees, or radians?

Revolutions

We seldom learn to use the simplest, nigh natural unit of measure for geometric angles, the
revolution
(rev). Other names for this unit are
full circle,
turn,
full turn, and
rotation
(rot). These are all good names and they all mean the same matter.

one full circle = 1 rev = 1 plow = 1 rot = 360°

Degrees

In elementary schoolhouse, nosotros learn that angles are measured in
degrees
(°).

ane full circumvolve = 360°

Radians

In high school trigonometry and calculus classes, nosotros learn that mathematicians adopt
radians
(rad).

1 total circle = 2π rad


Note:
If you take non yet learned about radians in school, you may ignore the radians in everything beneath.


Comparing Revolutions, Degrees, and Radians

Let’s compare revolutions and degrees (and radians).

words rev deg rad
no plough 0 0
quarter turn 1/4 ninety° π/2
half turn 1/two 180° π
3-quarter turn 3/4 270° 3π/2
full turn 1 360°
twelfth turn one/12 thirty° π/6
eighth turn 1/8 45° π/4
sixth turn one/6 60° π/iii
fifth turn 1/v 72° 2π/five
third turn 1/3 120° 2π/3
two turns 2 720°
three turns three 1080°

To convert from revolutions to degrees, multiply by 360. To convert from degrees to revolutions, divide by 360.

When you use degrees you are often working with integers, but when you utilise revolutions (or radians) you are often working with fractions (or decimals). Hand calculations are sometimes easier when you utilise revolutions just sometimes easier when yous utilise degrees. It’s good to know both ways.

Revolutions (turns) are a more rational and natural unit of measure out than degrees. You’ll become a deeper agreement of angles if you think about revolutions rather than degrees. An angle is more fundamentally a subdivision of a circle rather than a sum of degrees. For example, a right angle is more than fundamentally a quarter of a circle rather than a sum of 90 degrees.

Let’s separate the circle into n equal sectors (see diagram below). The angle of each sector is 1/n rev = 360/n° = 2π/n rad. It is easier to understand this if you think about revolutions rather than degrees (or radians).

Let’s look at some basic geometry using revolutions and degrees (and radians). The diagram below shows supplementary angles, complementary angles, and triangles. The concepts are clearer if yous recollect about revolutions rather than degrees. The arithmetic may be easier using degrees if you accept trouble adding and subtracting fractions.

Let’southward await at polygons (see diagram below). For a regular polygon with n sides, the exterior angle is 1/north rev = 360/n° = 2π/n rad. It is easier to understand this if you recall about revolutions rather than degrees (or radians). The interior angle is the supplement of the outside angle.


Teaching Revolutions, Degrees, and Radians

I think it would exist expert if teachers would introduce revolutions (turns) at the same time that they introduce degrees. This will help the students to understand angles at a more central level, less dependent on the arbitrary magic number 360. Teachers already innovate the general concept of revolutions (turns) when they say things similar “a full circumvolve is 360°”, but they tin can make the concept more numerically precise by saying “a full turn is 360°, a one-half plow is 180°, a quarter turn is 90°, and an 8th turn is 45°” or writing “1 rev = 360°, ane/ii rev = 180°, 1/4 rev = 90°, and 1/8 rev = 45°”. Students should occasionally do doing a few calculations using revolutions (turns) rather than degrees. Of course, students will need to spend most of their fourth dimension learning to calculate with degrees (and subsequently, radians), because that is the standard.


Angles in Trigonometry and Calculus

Finally, allow’due south accept a quick look at more advanced mathematics: trigonometry and calculus.

Nosotros tin consider using revolutions with trigonometric functions (sine, cosine, tangent). For example, instead of maxim cos(60°) = 1/2 or cos(π/3) = 1/ii using radians, we might want to say cos(1/half dozen) = 1/2 using revolutions. But this is not applied considering we depend on calculators to evaluate the trigonometric functions, and calculators typically have only DEG and RAD modes, non REV mode.

When nosotros get to deeper levels of mathematics, such every bit calculus and mathematical analysis, it turns out that radians are the most rational and natural units. For case, consider this primal equation: the limit of sin(10)/x as x approaches 0 is exactly 1. This equation would not exist then elegant if we used whatsoever unit other than radians.


Angles – The Revolution
©
20102016
Keith Enevoldsen

thinkzone.wlonk.com


What is 3 8 of a Full Rotation

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