# Can $\sin(\pi - \alpha)$ be written as $\sin(180^{\circ}-\alpha)$?

This is a simple question concerning $\sin(\pi - \alpha)$ when $\alpha$ is known. Is it correct to write it as $$\sin(180^{\circ} - \alpha),$$ as $\pi$ is $180^{\circ}$ in radians? For example, $\sin(\pi - 25^{\circ}) = \sin(155^{\circ}) \approx 0.423$, and the result is the ratio of the length of the opposite side to the length of the hypotenuse.

• I would say: $\quad \sin(\pi-\alpha_{[rad]})=\sin(180°-\alpha_{[°]})=\sin \alpha$ – georg Oct 30 '16 at 9:47
• Simply note that $^{\circ} \stackrel{\text{df}}{=} \dfrac{\pi}{180}$. As it is typographically awkward to write $^{\circ}$ by itself, we usually write it as $1^{\circ}$. – Berrick Caleb Fillmore Oct 30 '16 at 10:02

Yes it can be so written. If you say $\sin 2.5$ it is not clear whether the argument is in radians or in degrees. So better to say $\sin 2.5^0$ or $\sin 2.5^r$ The latter is not followed usually, it can be seen from the context.
But when referring to $\cos \pi/3$ it is obvious that it is in radians, no need to say $\cos 60^0.$
This doesn't have anything to do with sine itself but with conversion from degrees to radians and vice versa. Note that both degrees and radians are supposed to measure rotation and thus we must be able to add and subtract angles and this should correspond to addition and subtraction of real numbers (what I mean is that if we rotate by $\alpha$ and $\beta$ consecutively, this should be the same as rotating by $\alpha + \beta$). This is taken care of by addition formulas in trigonometry and isn't a concern to us at the moment. What is the concern is that this structure should be independent of whether we measure angles in radians or degrees, i.e. our conversion formula $x^\circ = x\frac{\pi }{180}\,\mathrm{rad}$ should preserve addition and this is immediately verified: $$(x+y)^\circ = (x+y)\frac{\pi }{180}\,\mathrm{rad} = x\frac{\pi }{180}\,\mathrm{rad}+y\frac{\pi }{180}\,\mathrm{rad} = x^\circ + y^\circ$$ This tells you that you can freely switch from radians to degrees, add and subtract them to your satisfaction, and then convert it back to radians. This is what I advise my students in high school to do when calculating, for example, $\sin\frac{7\pi}{12}$. It is not as easy to notice that $\frac{7\pi}{12} = \frac\pi 3 + \frac\pi 4$ as it is that $105^\circ = 60^\circ + 45^\circ$. One then applies addition formula for sine to conjure the result.
To contrast this, let me give you an example when things are not as simple as here. When converting degrees of celsius to degrees of fahrenheit you cannot just add temperatures expressed in one unit, and convert to the other. Trivial example would be $0^\circ C= 32^\circ F$. You could say that $0^\circ C + 0^\circ C= 0^\circ C$, but $32^\circ F + 32^\circ F = 64^\circ F$, a different result. There is a deeper mathematical reason why it is wrong to add temperatures in the first place, but it is beside the point here. All I wanted to demonstrate is that it is not always trivial to apply operations to measured quantities and expect the result is independent of the measuring system. This always requires verification.