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I am using a formula which calculates an angle $ θ_{h} $, and I am not sure if the result is (as I suspect) rad $\times$ degrees:

$$ θ_{h}=\dfrac{π}{2}\sin^{-1}\left(\dfrac{2}{π}-0.2782\right) = \dfrac{π}{2} \sin^{-1}\left(\dfrac{2-π\cdot0.2782}{π}\right) = \dfrac{π}{2}\sin^{-1}(0.3584) = \dfrac{π}{2} \cdot 21.0° $$

Now I think this is $π/2 \text{ rad } \times 21.0°$ degrees. So if I transform $π/2$ to $90°$, I can do: 90°+21°=111° $90°*21°=1890°$.

Based on Garry's comment, I re-calculate: $\dfrac{π}{2}\sin^{-1}(0.3584)= \dfrac{π}{2} \cdot 0.366_{rad}=0.575_{rad}=32.6°$. Right, I dont remember to have ever multiplied degrees with each other, that was a mistake.

  1. I can do this transformation right? It feels so odd becouse:

  2. If the above calculation/transformation I did is true, I think that when numbers are used to show an angle in rad, they should have a indication like Degrees do (∘), like so: $ (\dfrac{π}{2})_{θrad} $, so that one can know that this number is convertible to degrees (if needed) and it shows an angle. This would prevent mistakes like if someone instead of $ \dfrac{π}{2} $, wrote $ 1.57 $ (which is the result of $ \dfrac{3.14}{2} $), I might think that $ 1.57 $ is in degrees format! But $ 1.57_{θrad} $ makes more sense . Does something like this exist?

Christianidis Vasilis
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    Why do you think $\sin^{-1}$ gives you something in degrees and not radians? How did the product of two angles become the sum of them? – Gary Apr 28 '22 at 12:32
  • @Gary mistake fixed! – Christianidis Vasilis Apr 28 '22 at 12:45
  • My answer to your deleted comment: Put your calculator in radian mode. :) The final answer will be $\approx 0.5758 \text{ rad } \approx 33°$. – Gary Apr 28 '22 at 12:46
  • Radians are pure numbers. The ratio of circumference to radius, both measured in the same units, for example, centimeters or meters or (god forbid) inches where the units cancel. – MasB Apr 28 '22 at 12:58
  • @Gary you are right, I should not have multiplied degrees with each-other. But the questions remain. – Christianidis Vasilis Apr 28 '22 at 13:03
  • It's primarily the author's responsibility to write "$^\circ$" when degrees are intended. Of course, omitting the symbol is common (esp. here on Math.SE!). If there were a symbol for radians (I've occasionally seen people use a superscript $r$ or $c$), I suspect that omitting that symbol would be just as common. Consequently, it's also the reader's responsibility to stay vigilant. (Again, on Math.SE, when someone asks about an errant trig calculation, often the first pass at responding is: Is your calculator in the correct mode?) – Blue Apr 28 '22 at 13:16
  • There is no standard indicator for radians, but in physics, some authors use "rad" to indicate them. Radians are dimensionless, so there is no reason for the to be an indicator. Degrees are also not really used in mathematics or physics. – Angel Apr 28 '22 at 15:17

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