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$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to but isn't an angle), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(10313^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.

$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to but isn't an angle), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(233^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.

$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to an angle), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(10313^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.

$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to but isn't an angle), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(10313^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.

$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C}$), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(10313^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.

$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical quantities, each having both a numerical value and a unit.

An angle can be construed as a ratio of lengths: its number of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc that subtends the angle at a circle's centre}}{\text{radius of the circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.

It's instructive to understand the two standard versions of each trigonometric function as being in fact two different functions: one accepts an input with unit $^{\circ}$, the other accepts a unitless input (the $\textrm{rad}$ having been divided out so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to an angle), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

• $\sin(\pi)\neq\sin(180)=\sin(10313^{\circ}).$
• The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.
• Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.
• $$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$

Similarly, in the arc length formula $s=r\theta,$ the subtended angle is $\theta\textrm{ rad},$ not $\theta.$

• $“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$

The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.