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According to right triangle definition, trigonometric functions relates the angle to the ratio of sides of triangle. These functions take angles as input.

According to unit circle definition, trigonometric functions gives the coordinates of the points where the terminal side of the angle intersects the unit circle. Here, also trigonometric functions take angles (even it is in radians still it is not just a number).

According to both definitions, trigonometric functions are the functions of angles. Then, how can a trigonometric function take just a real number as input?

What does it even mean?

Many websites (even my textbook) point that radians and real numbers are just same. But I can't understand how can radians act as just numbers. Yes, it is dimensionless quantity. But it represents the quantity of something so it can't be just numbers.

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    $\begingroup$ It really can be "just numbers". $\sin(x)$ means sine of $x$ radians, or more generally, the extension of the angle-defined function sine to more general inputs $\endgroup$
    – FShrike
    Commented Aug 18, 2021 at 9:45
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    $\begingroup$ Why can't a real number represent quantity of something. "You have 5 toffees". Doesn't $5$ represent quantity here? In case of angles, an angle of $45°$ =$\frac{\pi}{4}=0.785$ represent the angle subtended by an arc with length $0.785$ times its radius at the centre of the circle of which it is a part. Don't you think the number $0.785$ is enough to give the quantity of this angle? $\endgroup$
    – Aman Kushwaha
    Commented Aug 18, 2021 at 9:56
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    $\begingroup$ Ratios between lengths are also quantities and they do not need a unit. $\sin(x)$ is basically such a ratio. $\endgroup$
    – Peter
    Commented Aug 18, 2021 at 10:02
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    $\begingroup$ The argument to a trigonometric function is a ratio as well; it is the ratio of the length of the arc along a circle and the radius of the circle. By dealing only with unit circles, one can lose sight of that. $\endgroup$
    – robjohn ♦
    Commented Aug 18, 2021 at 10:08
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    $\begingroup$ @VedantChhapariya Do you really think writing "radians" after $5$ makes any difference to the angle. $5$ is just a ratio of two same dimensional quantity and hence the number $5$ is enough to tell you the angle, unlike some other quantity like length where "cm" makes sense because otherwise a length of $5$ would be ambiguous as there is no such definition of a number being a length. But in case of angles you have. $\endgroup$
    – Aman Kushwaha
    Commented Aug 18, 2021 at 10:08

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