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supermiedos
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What differente does it make? As far as I can see it, the limit definition of a derivative shouldn't be affected by the fact that x is expressed in radians or degrees...
Why derivative of sin x is cos x, only if x is in radians?
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In summary, the limit definition of the derivative is not affected by the choice of radians or degrees as long as the limit is taken in radians. This is because the radian has two meanings, one as an angle and one as a length, making it a convenient unit for calculus. Additionally, the chain rule can be used to derive the derivative for sine in degrees by converting the argument to radians. However, the limit that is essential for differentiating sin(x) only holds when x is in radians.
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agentdoggett
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supermiedos said:
The limit definition of the derivative relies heavily on approximations to length. The radian has 2 meanings, that of an angle and that of a length so calculus works very nicely when angles are represented in radians.
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Infrared
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You can think of it mechanically in terms of the chain rule. If the nice derivative rule holds for radians, it can't hold for an angle measure that differs from radians by a factor.
If f(y)=siny, where y is in degrees, then f(y)=sin(yπ/180), where y is in degrees but the argument of the sine is now in radians so we know how to take its derivative. This has a derivative of π/180*cos(πy/180). The logic here is the same as why, for example [itex] \frac {d}{dx} e^x = e^x [/itex] but [itex] \frac {d}{dx} e^{2x} \neq e^{2x} [/itex]
If f(y)=siny, where y is in degrees, then f(y)=sin(yπ/180), where y is in degrees but the argument of the sine is now in radians so we know how to take its derivative. This has a derivative of π/180*cos(πy/180). The logic here is the same as why, for example [itex] \frac {d}{dx} e^x = e^x [/itex] but [itex] \frac {d}{dx} e^{2x} \neq e^{2x} [/itex]
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Bipolarity
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The limit [itex] \lim_{x→0}\frac{sin x}{x} = 1 [/itex] holds only when x is in radians. This limit is essential to differentiating sin(x) from first principles.
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supermiedos
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Thank you so much for your answers, now I see it clearly :)
1. Why is the derivative of sin x equal to cos x only if x is in radians?
The derivative of a function represents the rate of change of that function. In the case of sin x, the derivative is equal to cos x. This is true only if x is measured in radians because the derivative of sin x is defined using the unit circle, which is based on radians. If x is measured in degrees, the derivative of sin x would be different.
2. What is the relationship between radians and the unit circle?
Radians are a unit of measurement for angles, just like degrees. However, radians are based on the radius of a circle, while degrees are based on dividing a circle into 360 equal parts. The unit circle is a circle with a radius of 1, and its circumference is divided into 2π radians. This relationship is crucial in understanding why the derivative of sin x is cos x only if x is in radians.
3. Can the derivative of sin x be calculated using other units of measurement?
Yes, the derivative of sin x can be calculated using units other than radians, such as degrees or gradients. However, the formula for calculating the derivative will be different for each unit of measurement. For example, if x is measured in degrees, the derivative of sin x would be equal to cos x multiplied by π/180.
4. Why do we use radians instead of degrees in calculus?
In calculus, radians are the preferred unit of measurement for angles because they are directly related to the derivative and the concept of instantaneous rate of change. Radians also make certain calculations, such as the derivative of trigonometric functions, simpler and more intuitive. Additionally, many mathematical formulas and equations are defined using radians as the unit of measurement.
5. Can the derivative of sin x be calculated using calculus if x is not measured in radians?
Yes, the derivative of sin x can be calculated using calculus even if x is not measured in radians. However, the formula for calculating the derivative will be different, depending on the unit of measurement used. It is essential to use the correct formula for the specific unit of measurement to get an accurate result.
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