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I recently started $11^{th}$ grade Trigonometry.

Prior to this, I learnt about the three general systems of angle measurement, mainly the Circular System of Angles, dealing with radians

The very first section in the first chapter related to Trigonometry was regarding Trigonometric Functions of Real Numbers.

Shouldn't Trigonometric Functions have angles as inputs, in degrees, grades, radians or some other angle unit.

I did read about the relationship between real numbers and radian measures. I think I misunderstood some part of it.

It doesn't mean that $x^c = x$, right? In my opinion, it means that if $x^c$ is some angle, then $x \in \Bbb R$. Which one of the above mentioned assumptions is right?

I also am confused about the term radian measure. In my opinion, it means that if $\theta = x^c$, then $x$ is called the radian measure of $\theta$. Am I right here as well?

If I'm right in both of these cases, does the term trigonometric functions of real numbers mean that $sin$ $x$ is a way of saying $sin(x^c)$, since the superscript $^c$ is usually omitted while writing radians?

This might sound like a simple question but its really confusing for me. Please let me know about any misconceptions I might have about radians and also what the right thing is.

Thanks!

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2 Answers 2

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Radians and degrees are two different units of angle measure. Like we measure length in metres or time in seconds, we measure angles in radians. Trigonometric functions do take angles as inputs in both units. So, $\sin 60^o , \sin \frac{\pi}{3}$ are both equivalent, just as $100\ cm$ is equivalent to a meter.

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  • $\begingroup$ Thanks a lot, so basically, when we write $sin \dfrac{\pi}{3}$, we mean $sin \Big ( \dfrac {\pi}{3} \Big )^c = sin(60^o) = \dfrac{\sqrt{3}}{2}$ and for any angle $x^c$, radian measure does mean $x$, and the relationship between real number and radian measures tells that if $x^c$ is any angle, then $x \in \Bbb R$, right? $\endgroup$ Commented Apr 25, 2020 at 18:40
  • $\begingroup$ @Rajdeep_Sindhu Yes, exactly. $\endgroup$ Commented Apr 25, 2020 at 18:42
  • $\begingroup$ You have no idea how confused I was earlier and how thankful I am now! $\endgroup$ Commented Apr 25, 2020 at 18:42
  • $\begingroup$ That’s good for you! $\endgroup$ Commented Apr 25, 2020 at 19:00
  • $\begingroup$ I'm confused again. Didn't quite get @J.G.'s answer. Please check out the comment and let me know if you can explain it to me. $\endgroup$ Commented Apr 25, 2020 at 19:03
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An arc of a radius-$r$ circle subtending an angle $\theta$ at its centre has length $s=kr\theta$ for some constant $k$, which depends on the units in which you measure angles. Regardless of that choice, $\theta$ being a dimensionless number implies $k$ is too. If we work with degrees, $k=\frac{\pi}{180}$; if we work with radians, $k=1$, which is the motive for using radians. So now $s=r\theta$, angles are dimensionless, and "$1$ radian" is just a fancy name for $1$. Why we even mention radians given this fact is another question, which I've addressed before. But if you bear this point in mind, all your concerns evaporate. For example, $\sin0.3$ is synonymous with $\sin(0.3\,\text{radians})$, because $0.3$ is synonymous with $0.3\,\text{radians}$.

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  • $\begingroup$ The answer that you linked stated that a quantity being dimensionless doesn't mean that it is unit-less. So, what does it mean for a quantity to be dimensionless? What exactly is meant by dimensions here? $\endgroup$ Commented Apr 25, 2020 at 18:48
  • $\begingroup$ So, when we say : trigonometric functions of real numbers, we mean trigonometric functions of angles, right? But, how? I just can't get it. Also, if 1 radian is just a fancy name for 1, what is the points of units? Pardon me if I'm being annoying, I'm just so confused... $\endgroup$ Commented Apr 25, 2020 at 18:50
  • $\begingroup$ @Rajdeep_Sindhu I think what J.G. means is that its usage is not super strict. Writing $\sin 1$ would be a shorter and more relaxed way of writing what would actually mean $\sin (1 \, radian)$. $\endgroup$ Commented Apr 25, 2020 at 19:10
  • $\begingroup$ Oh, that makes sense, let's wait till he confirms it... $\endgroup$ Commented Apr 25, 2020 at 19:11
  • $\begingroup$ @Rajdeep_Sindhu You know how rectangles have area $A=hw$? Here $h$ and $w$ each have the dimension of length, say $L$, whereas $A$ has dimension $L^2$. You can work out the dimension of any one variable from the other two, e.g. $[h]=[A]/[w]=L^2/L=L$. In some cases all exponents would be $0$, so the quantity is dimensionless, like $\theta=s/r$. But it makes sense to have units sometimes, because (as per my link) you don't want to confuse angles with solid angles or LD50s. $\endgroup$ Commented Apr 25, 2020 at 19:15

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