What does radian mode do? (1 Viewer)
- Thread starter sadpwner
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http://mathwithbaddrawings.com/2013/05/02/degrees-vs-radians/
i, Liz.
In degree mode, the calculator is essentially replacing the sine
function with a different function whose argument is a number of
degrees. If we call this new function sind, we can define it as
sind(x) = sin(pi/180 x)
since to calculate it we have to convert x degrees to radians and then
take the actual sine.
Take the derivative of this with respect to x, and you get
d/dx sind(x) = pi/180 cos(pi/180 x)
= pi/180 cosd(x)
Of course, what the calculator calls the cosine is now really this
"cosd" function.
Does that help? This also explains why we use radians when we do
calculus. Like many things mathematicians do, this makes everything
easier.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
http://mathforum.org/library/drmath/view/53779.html
http://www.purplemath.com/modules/radians.htm
Radians
Why do we have to learn radians, when we already have perfectly good degrees? Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc).
The 360° for one revolution ("once around") is messy enough. Why is the value for one revolution in radians the irrational value 2π? Because this value makes the math work out right. You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to make sense, the "numerical" value corresponding to 360° needed to be defined as (that is, needed to be invented having the property of) "2π is the numerical value of 'once around'."
i, Liz.
In degree mode, the calculator is essentially replacing the sine
function with a different function whose argument is a number of
degrees. If we call this new function sind, we can define it as
sind(x) = sin(pi/180 x)
since to calculate it we have to convert x degrees to radians and then
take the actual sine.
Take the derivative of this with respect to x, and you get
d/dx sind(x) = pi/180 cos(pi/180 x)
= pi/180 cosd(x)
Of course, what the calculator calls the cosine is now really this
"cosd" function.
Does that help? This also explains why we use radians when we do
calculus. Like many things mathematicians do, this makes everything
easier.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
http://mathforum.org/library/drmath/view/53779.html
http://www.purplemath.com/modules/radians.htm
Radians
Why do we have to learn radians, when we already have perfectly good degrees? Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, "83%" has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc).
The 360° for one revolution ("once around") is messy enough. Why is the value for one revolution in radians the irrational value 2π? Because this value makes the math work out right. You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you'll learn later, mathematicians like to work with the "unit" circle, being the circle with r = 1. For the math to make sense, the "numerical" value corresponding to 360° needed to be defined as (that is, needed to be invented having the property of) "2π is the numerical value of 'once around'."
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everal advantages to radians:
1) Conversion from angular displacement to linear displacement is much easier. For example, if a wheel with radius 1 foot turns 50 radians, the wheel will have traveled 50 feet.
2) The definition is less arbitrary. A radian is just one radius' worth of distance on the edge of the circle, whereas a degree is one one-hundred-eightieth of a full rotation. The only less arbitrary unit I could think of would be rotations (though it lacks some of the other nice features of radians).
3) Trigonometric functions are useful for things that do not involve angles of any sort. When you're using trigonometric functions to deal with waves, degrees and radians don't really have any meaning. However, the same familiar trigonometric function that maps an angle in radians to the length of the opposite side of a unit-hypotenuse triangle (this is the one you get if you hit "sin" on your calculator in radian mode, which is what mathematicians mean when they say "sine") also describes a variety of other useful situations. The one that maps an angle in degrees to the length of the opposite side (this is the one you get if you hit "sin" on your calculator in degree mode, hereafter referred to as "degree-sine") requires a little more tweaking to do the same work.
4) Calculus. The derivative of the sine function is the cosine function. The derivative of the degree-sine function is not the degree-cosine function, but rather a (pi/180) times the degree-cosine function. Be thankful that you don't have to deal with these kind of fudge factors.
5) Complex numbers. If you take a complex number z and write its position on the complex plane in polar coordinates (r,theta) (using radians, of course!), then z is actually equal to r*e^(i*theta). If you want to pull of the same trick with degrees, it'll cost you some more fudge factors.
Advantages to degrees:
1) You're already used to it.
2) It's used a lot for measuring angles (though not as often for doing the many other jobs that trigonometry is useful for).
3) There are a whole number of degrees in a full rotation (but really...180? I mean it does have a lot of factors, but talk about arbitrary. At least they could have gone with metric degrees with 100, 10, or even 1 in a rotation or something.)
http://ask.metafilter.com/85719/Why-do-we-use-radians
1) Conversion from angular displacement to linear displacement is much easier. For example, if a wheel with radius 1 foot turns 50 radians, the wheel will have traveled 50 feet.
2) The definition is less arbitrary. A radian is just one radius' worth of distance on the edge of the circle, whereas a degree is one one-hundred-eightieth of a full rotation. The only less arbitrary unit I could think of would be rotations (though it lacks some of the other nice features of radians).
3) Trigonometric functions are useful for things that do not involve angles of any sort. When you're using trigonometric functions to deal with waves, degrees and radians don't really have any meaning. However, the same familiar trigonometric function that maps an angle in radians to the length of the opposite side of a unit-hypotenuse triangle (this is the one you get if you hit "sin" on your calculator in radian mode, which is what mathematicians mean when they say "sine") also describes a variety of other useful situations. The one that maps an angle in degrees to the length of the opposite side (this is the one you get if you hit "sin" on your calculator in degree mode, hereafter referred to as "degree-sine") requires a little more tweaking to do the same work.
4) Calculus. The derivative of the sine function is the cosine function. The derivative of the degree-sine function is not the degree-cosine function, but rather a (pi/180) times the degree-cosine function. Be thankful that you don't have to deal with these kind of fudge factors.
5) Complex numbers. If you take a complex number z and write its position on the complex plane in polar coordinates (r,theta) (using radians, of course!), then z is actually equal to r*e^(i*theta). If you want to pull of the same trick with degrees, it'll cost you some more fudge factors.
Advantages to degrees:
1) You're already used to it.
2) It's used a lot for measuring angles (though not as often for doing the many other jobs that trigonometry is useful for).
3) There are a whole number of degrees in a full rotation (but really...180? I mean it does have a lot of factors, but talk about arbitrary. At least they could have gone with metric degrees with 100, 10, or even 1 in a rotation or something.)
http://ask.metafilter.com/85719/Why-do-we-use-radians
Martin_SSEDU
Active Member
You need to use radian mode for most Trig questions or else you won't get the marks, pretty much the only reason you'll need to ever know for HSC... I lost many many silly marks to not having my calc in radian mode. Just don't forget to change back to degree mode when your done.