Is an angle a dimensionless measure without unit?

Question

Is there a difference of unit between dimensionless measures of angles and lenght in triangles? For example two sides are composed of a distance $0.85 cm+0.4cm=1.25 cm$ and at the same time $0.4cm=\cos\theta =0.4$ without a unit, and the three altitudes are situated on each sides of the triangle, and the base is $1cm$? And beetween $0.5cm$ and $0.5cm$ is found the altitude of the base $1 cm$.Between $0.85cm $and $0.4 cm$ is situated an altitude.

For consecutive numbers or non consecutive numbers $x<y<z$ I have the following answer:

$\sqrt\frac{(z-y)}{z}=\cos B$

$\sqrt\frac{y}{z}=\sin B$

$\frac{x}{z}=\cos C$

$((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin C$

$(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times((1-\frac{x}{z})\times\sqrt\frac{(x+z)}{(z-x)})=\sin A$

$(\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})-(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times(\frac{x}{z})=\cos A$

And all I have to do is replace $x<y<z$ by any real numbers bigger than zero.

$((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\sin C)=\sin A$

$(\frac{\sin B}{\sin C})-((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\cos C)=\cos A$

$((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\sin C)=\sin B$

$(\frac{\sin A}{\sin C})-((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\cos C)=\cos B$

$ ((((\frac{\sin B}{\sin A})\times\cos A)+\cos B)\times\sin A) = \sin C, $

$ (\frac{\sin B}{\sin A})-((((\frac{\sin B}{\sin A})\times\cos A)+\cos B)\times\cos A) = \cos C. $

And you can obtain all six angles $\alpha \beta \gamma$ of trigonometric functions $\sin\theta$ and $\cos\theta$ with consecutive or non consecutive numbers, where $x<y<z$ are real numbers.

Answer 1

First of all, angles are dimensionless measures. However, they have to be quantified, in degrees or radians as appropriate.

I do not understand your numerical example. You could be thinking about the sine and cosine rules. Do read up more about it.

Answer 2

I would regard angles in radians as dimensionless. From Radian at Wikipedia

"Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc."

Hence a length divided by a length. Whether you use metres, feet, or cubits for the lengths, you will get the same result.

It is common to indicate radians, as if they were a unit, but this is mainly because degrees have been used historically. Within pure mathematics, this is often not done as radians are usual.

Answer 3

Angles can be treated as dimensioned objects. Dimensions do not exist in nature but are an artefact of the forumlae.

Angle-units, such as degrees, should be always given, but even something like SI can not make its mind up as to which angle is mathematically unit.

In the case where one is working with radii, and functions derived from taylor-series, the radian is the meaning of unity. According to the nature of the geometric product, then square radians, or rhombic radians are the measure.

In the case where one is working with all-space as one (eg Gauss's flux law or cycles against time), then the whole space is the unit. This is coherent to the prism-product of figures.

The difference between the two cases is the matter of rationalisation of formulae between the CGS and SI sets of formulae.