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Find an angle α that is coterminal with an angle measuring 870°, where α < 360° .

α = 150 °

Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°.

  1. Add 360° to the given angle.
  2. If the result is still less than 0°, add 360° again until the result is between 0° and 360°.
  3. The resulting angle is coterminal with the original angle.

Finding an angle coterminal with an angle measuring less than 0°

Show the angle with measure −45° on a circle and find a positive coterminal angle α such that 0° ≤ α <360°.

Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle.

Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°:

45° + 360° = 315°

We can then show the angle on a circle, as in [link] .

A graph showing the equivalence of a 315 degree angle and a negative 45 degree angle.

Find an angle β that is coterminal with an angle measuring −300° such that β < 360° .

β = 60 °

Finding coterminal angles measured in radians

We can find coterminal angles    measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

Given an angle greater than 2 π , find a coterminal angle between 0 and 2 π .

  1. Subtract 2 π from the given angle.
  2. If the result is still greater than 2 π , subtract 2 π again until the result is between 0 and 2 π .
  3. The resulting angle is coterminal with the original angle.

Finding coterminal angles using radians

Find an angle β that is coterminal with 19 π 4 , where 0 β < 2 π .

When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of 2 π radians:

19 π 4 2 π = 19 π 4 8 π 4 = 11 π 4

The angle 11 π 4 is coterminal, but not less than 2 π , so we subtract another rotation:

11 π 4 2 π = 11 π 4 8 π 4 = 3 π 4

The angle 3 π 4 is coterminal with 19 π 4 , as shown in [link] .

A graph showing a circle and the equivalence between angles of 3pi/4 radians and 19pi/4 radians.

Find an angle of measure θ that is coterminal with an angle of measure 17 π 6 where 0 θ < 2 π .

7 π 6

Determining the length of an arc

Recall that the radian measure     θ of an angle was defined as the ratio of the arc length     s of a circular arc to the radius r of the circle, θ = s r . From this relationship, we can find arc length along a circle, given an angle.

Arc length on a circle

In a circle of radius r , the length of an arc s subtended by an angle with measure θ in radians, shown in [link] , is

s = r θ
Illustration of circle with angle theta, radius r, and arc with length s.

Given a circle of radius r , calculate the length s of the arc subtended by a given angle of measure θ .

  1. If necessary, convert θ to radians.
  2. Multiply the radius r by the radian measure of θ : s = r θ .

Finding the length of an arc

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

  1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
  2. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
  1. Let’s begin by finding the circumference of Mercury’s orbit.
    C = 2 π r = 2 π ( 36  million miles ) 226  million miles

    Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:

    ( 0.0114 ) 226  million miles = 2 .58 million miles
  2. Now, we convert to radians:
    radian = arclength radius = 2. 58 million miles 36  million miles = 0.0717

Questions & Answers

what is variations in raman spectra for nanomaterials
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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