Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
They both, of course, are arbitrary ways to measure angle, but one turns out to me more convenient for, say, calculus applications because it plays nicer with other functions that are useful.
(There is nothing inherent in a circle that says its total angle measure has to be 2*pi radians, although you can make the case that it is inherent in a circle that the ratio of the circumference to the diameter of a circle is constant, and that we might want to give this constant a name (that is, (2 pi r) / (2 r) = pi for any circle), and that it might (or might not) be useful to use this quantity to define angle measure.)
example problem
-5(pie)over 6
you multiply the problem by 180 over (pie) both of the (pies) cancel out. and your answer should be -150 degrees.
Citations
"Why Do We Use Radians? | Ask MetaFilter." Ask MetaFilter | Community Weblog. Web. 21 Sept. 2010. .
Yep its Taylor answering her prompt a day early...
Radians are important in trig. for many reasons. One of them is that if you are looking for the exact value of degrees you would use radians (PI/2, PI/3, PI/4, and PI/6) to help you find the exact value of the measurement. Another important use of radians in trig is that it is needed for the equations that have THETA or the central angle of an object. Radian also 'specify an angle by measuring the length around the path of a unit circle'(http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians).
If you want to convert degrees to radians you would multiply the degrees times PI/180, and solve.
Example problem:
Convert 450 Degrees to radians
450 degrees x Pi/180 First, you will set up ur problem.
450PI/180 Then you would multiply 450 degrees and PI/180 or place 450 degrees over 180 and get this. Remember PI is considered a variable.
5PI/2 Lastly, you would simplify the fraction or divide and you will get this number as your answer. That is how you convert degrees to radians!
(Don't know how to site so.. i will just give you the site that I went to to find the information that I do not own or take credit for!!!)
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them.
Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
Citations http://en.wikipedia.org/wiki/Radian
example 90 degrees
you multiply 90 x pi/180 then you divide by 180 and you will get a decimal and you just go on the calculator and hit the math key and hit fraction.
This is Nathan with the comment on the Week 4 Blog prompt.
We use radians in trignometric functions which are functions of an angle. They are used to relate the angles of a triganle to the lengths of the sides of the triangle. Radians specify an angle by measuring the length around a path of the unit circle and constitute a special argument to the sin and cosine functions. In particular, only sines and cosines that map radians to ratio satisfy the differential equations that classically describe them. Radians are also important because they are needed for equations that have theta(central angle) in it.
Ex. Convert 250 degrees to radians. 250 x π/180=250π/180=25π/180 All you do, is multiply the angle by π/180.
Ex.2 Convert 150 degrees to radians. 150 x π/180=150π/180=5π/6
Now, here's an example of radians to degrees. 14π x 180/π=2520 degrees. All you do here, is flip the equations around. You multiply by 180/π to get from radians to degrees.
It is important to use radians in trig because they are use to say what the angles and lenghts of a triangle are and how they relate. Radians measure the length of a path around the unit circle. They are also used with sin and cos. They are also important when dealing with equations that have theta or a central angle in it.
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.
To convert from degrees to radians you just multiply pi/180
The reason why it's important to use radians in trig because it tells you the angle by measuring the length around the unit circle and constitute a special argument to the sine and cosine functions. Only sine and cosine that can map radians to ratios differential equations that can classically describe them. It is also important for studying triangles and modeling periodic phenomena.
To convert from degrees to radians you just multiply PI/180
Example:
30 degrees to radians
30 X PI/180= 30Pi/180= PI/6
320 degrees to radians
320 X PI/180= 320PI/180= 16PI/9
And if you ever want to change radians to degrees back you would just multiply 180/PI.
Examples:
16PI/9 convert to degrees
16PI/9 x 180/PI= 320 degrees
PI/6 convert to degrees
PI/6 x 180/PI= 30 degrees
( This was the easiest thing i've learned all year)
“When it comes to differentiating, polar coordinates are more convenient because a derivative gives the slope of the tangent to a curve at some specific point. Since a curve is round, it makes more sense to use a circular approximation than a rectangular approximation, for sake of accuracy and precision. This doesn't make too much of a difference in terms of derivatives, but once you start hitting integrals, differentials, partials derivatives, and gradients of multivariable functions, you'll definitely want to be using radians instead of degrees.
Also, radians are generally preferred because they are all related by a factor of pi, a natural number, instead of 180 degrees, an arbitrary number, which makes them easier to calculate without a calculator (than degrees).”
To convert degrees to radians, you must take that degree and multiply it by pi/180
EXAMPLE: Convert 18 degrees to radians. 18 x pi/180 = 10pi
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency.
Also, radians are generally preferred because they are all related by a factor of pi, a natural number, instead of 180 degrees, an arbitrary number, which makes them easier to calculate without a calculator.”
To convert degrees to radians, you must take that degree and multiply it by pi/180 example:
Convert 200° into radian measure: 200° (pie/180°) = 200/180 pie radians
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them.
Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
Additionally all of the formulas given in trig require that the angle be in radians for the formula to be applied. Thus there is a certain practicality to it.
Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
ReplyDeleteThey both, of course, are arbitrary ways to measure angle, but one turns out to me more convenient for, say, calculus applications because it plays nicer with other functions that are useful.
(There is nothing inherent in a circle that says its total angle measure has to be 2*pi radians, although you can make the case that it is inherent in a circle that the ratio of the circumference to the diameter of a circle is constant, and that we might want to give this constant a name (that is, (2 pi r) / (2 r) = pi for any circle), and that it might (or might not) be useful to use this quantity to define angle measure.)
example problem
-5(pie)over 6
you multiply the problem by 180 over (pie)
both of the (pies) cancel out.
and your answer should be -150 degrees.
Citations
"Why Do We Use Radians? | Ask MetaFilter." Ask MetaFilter | Community Weblog. Web. 21 Sept. 2010. .
Yep its Taylor answering her prompt a day early...
ReplyDeleteRadians are important in trig. for many reasons. One of them is that if you are looking for the exact value of degrees you would use radians (PI/2, PI/3, PI/4, and PI/6) to help you find the exact value of the measurement. Another important use of radians in trig is that it is needed for the equations that have THETA or the central angle of an object. Radian also 'specify an angle by measuring the length around the path of a unit circle'(http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians).
If you want to convert degrees to radians you would multiply the degrees times PI/180, and solve.
Example problem:
Convert 450 Degrees to radians
450 degrees x Pi/180 First, you will set up ur problem.
450PI/180 Then you would multiply 450 degrees and PI/180 or place 450 degrees over 180 and get this. Remember PI is considered a variable.
5PI/2 Lastly, you would simplify the fraction or divide and you will get this number as your answer. That is how you convert degrees to radians!
(Don't know how to site so.. i will just give you the site that I went to to find the information that I do not own or take credit for!!!)
http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
ReplyDeleteRadians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them.
Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
Citations
http://en.wikipedia.org/wiki/Radian
example
90 degrees
you multiply 90 x pi/180
then you divide by 180 and you will get a decimal and you just go on the calculator and hit the math key and hit fraction.
answer should be pi/2
This is Nathan with the comment on the Week 4 Blog prompt.
ReplyDeleteWe use radians in trignometric functions which are functions of an angle. They are used to relate the angles of a triganle to the lengths of the sides of the triangle.
Radians specify an angle by measuring the length around a path of the unit circle and constitute a special argument to the sin and cosine functions. In particular, only sines and cosines that map radians to ratio satisfy the differential equations that classically describe them. Radians are also important because they are needed for equations that have theta(central angle) in it.
Ex. Convert 250 degrees to radians.
250 x π/180=250π/180=25π/180
All you do, is multiply the angle by π/180.
Ex.2 Convert 150 degrees to radians.
150 x π/180=150π/180=5π/6
Now, here's an example of radians to degrees.
14π x 180/π=2520 degrees.
All you do here, is flip the equations around. You multiply by 180/π to get from radians to degrees.
Citations
http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
It is important to use radians in trig because they are use to say what the angles and lenghts of a triangle are and how they relate. Radians measure the length of a path around the unit circle. They are also used with sin and cos. They are also important when dealing with
ReplyDeleteequations that have theta or a central angle in it.
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.
To convert from degrees to radians you just multiply pi/180
Example:
Convert 20degrees to radians
20 x pi/180 = 20pi/180 = pi/9
http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
http://en.wikipedia.org/wiki/Radian
The reason why it's important to use radians in trig because it tells you the angle by measuring the length around the unit circle and constitute a special argument to the sine and cosine functions. Only sine and cosine that can map radians to ratios differential equations that can classically describe them. It is also important for studying triangles and modeling periodic phenomena.
ReplyDeleteTo convert from degrees to radians you just multiply PI/180
Example:
30 degrees to radians
30 X PI/180= 30Pi/180= PI/6
320 degrees to radians
320 X PI/180= 320PI/180= 16PI/9
And if you ever want to change radians to degrees back you would just multiply 180/PI.
Examples:
16PI/9 convert to degrees
16PI/9 x 180/PI= 320 degrees
PI/6 convert to degrees
PI/6 x 180/PI= 30 degrees
( This was the easiest thing i've learned all year)
Cite: http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
http://www.experiencefestival.com/trigonometric_function_-_the_significance_of_radians
“When it comes to differentiating, polar coordinates are more convenient because a derivative gives the slope of the tangent to a curve at some specific point. Since a curve is round, it makes more sense to use a circular approximation than a rectangular approximation, for sake of accuracy and precision. This doesn't make too much of a difference in terms of derivatives, but once you start hitting integrals, differentials, partials derivatives, and gradients of multivariable functions, you'll definitely want to be using radians instead of degrees.
ReplyDeleteAlso, radians are generally preferred because they are all related by a factor of pi, a natural number, instead of 180 degrees, an arbitrary number, which makes them easier to calculate without a calculator (than degrees).”
To convert degrees to radians, you must take that degree and multiply it by pi/180
EXAMPLE: Convert 18 degrees to radians.
18 x pi/180
= 10pi
http://wiki.answers.com/Q/Why_use_radians_when_differentiating_trig_functions
Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency.
ReplyDeleteAlso, radians are generally preferred because they are all related by a factor of pi, a natural number, instead of 180 degrees, an arbitrary number, which makes them easier to calculate without a calculator.”
To convert degrees to radians, you must take that degree and multiply it by pi/180
example:
Convert 200° into radian measure:
200° (pie/180°) = 200/180 pie radians
cite:
http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
The reason to use radian measurement instead of degree measurement is that radian measurment gives us a unit of measure that is consinant with a ruler and distance. Degrees do not give us distance.
ReplyDeleteRadians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them.
Radians are convenient because we have defined trigonometric functions in a certain way, and using these trigonometric functions is convenient. Because of the way that sine and cosine are defined, if you feed in to the sine function an angle measure whose units are degrees, you get out an ugly answer (or at least one that has a factor of pi/180 that you have to keep track of), whereas if you feed the sine function an angle measure with units of radians, you get out (comparatively) nicer answers.
http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians
I know this was used by another student, but it really makes sense, its an easier notation and is convenient.
To convert to radians
degrees X Pi/180
Ex. 90 degrees
90(π/180)
you get 90π/180
which simplifies to π/2
yay!
Additionally all of the formulas given in trig require that the angle be in radians for the formula to be applied. Thus there is a certain practicality to it.
ReplyDelete