<< Chapter < Page Chapter >> Page >

Converting degrees to radians

Convert 15 degrees to radians.

In this example, we start with degrees and want radians, so we again set up a proportion, but we substitute the given information into a different part of the proportion.

θ 180 = θ R π 15 180 = θ R π 15 π 180 = θ R π 12 = θ R
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Convert 126° to radians.

7 π 10

Got questions? Get instant answers now!

Finding coterminal angles

Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of to 360° , or 0 to 2 π . It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.

It is possible for more than one angle to have the same terminal side. Look at [link] . The angle of 140° is a positive angle, measured counterclockwise. The angle of –220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles    . Every angle greater than 360° or less than is coterminal with an angle between and 360° , and it is often more convenient to find the coterminal angle within the range of to 360° than to work with an angle that is outside that range.

A graph showing the equivalence between a 140 degree angle and a negative 220 degree angle.  The 140 degrees angle is a counterclockwise rotation where the 220 degree angle is a clockwise rotation.
An angle of 140° and an angle of –220° are coterminal angles.

Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it—the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260° .

An angle’s reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See [link] for examples of reference angles for angles in different quadrants.

Four side-by-side graphs. First graph shows an angle of t in quadrant 1 in its normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.

Coterminal and reference angles

Coterminal angles are two angles in standard position that have the same terminal side.

An angle’s reference angle    is the size of the smallest acute angle, t , formed by the terminal side of the angle t and the horizontal axis.

Given an angle greater than 360° , find a coterminal angle between and 360°

  1. Subtract 360° from the given angle.
  2. If the result is still greater than 360° , subtract 360° again till the result is between and 360° .
  3. The resulting angle is coterminal with the original angle.

Finding an angle coterminal with an angle of measure greater than 360°

Find the least positive angle θ that is coterminal with an angle measuring 800° , where θ < 360 ° .

An angle with measure 800° is coterminal with an angle with measure 800 360 = 440° , but 440° is still greater than 360° , so we subtract 360° again to find another coterminal angle: 440 360 = 80° .

The angle θ = 80° is coterminal with 800° . To put it another way, 800° equals 80° plus two full rotations, as shown in [link] .

A graph showing the equivalence between an 80-degree angle and an 800-degree angle where the 800 degree angle is two full rotations and has the same terminal side position as the 80 degree.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask