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Why is $\sin(30^\circ)$ exactly $0.5,$ when it could be 0.49999 or something else? There must be an easy geometric explanation?

amWhy
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Rodrigo
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    Hint: Draw an equilateral triangle and a perpendicular bisector. – JimmyK4542 Jun 09 '14 at 21:05
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    The perpendicular bisector divides the 60 degrees angle in two 30 degrees. So what? Sorry, I didn't get it. – Rodrigo Jun 09 '14 at 21:13
  • So what? So then you'll have the same theorem I mention in my answer...this , in fact, is its proof, but of course: one also need to know that a perp. bisector of the angle oppositve the base in an equilateral triangle is also the base's median. I can't really see how to get rid, in the most basic cases, of euclidean geometry. – DonAntonio Jun 09 '14 at 21:17
  • I was using the length of the bisector as 1, thus my confusion. Thank you all! – Rodrigo Jun 09 '14 at 21:25

4 Answers4

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Draw an equilateral triangle $ABC$ with $AB = BC = CA = 1$. Let $D$ be the midpoint of $BC$.

Now, $AD$ is the perpendicular bisector of $BC$, so $BD = CD = \tfrac{1}{2}$ and $\angle ADB = 90^{\circ}$.

Also, $AD$ is the angle bisector of $\angle BAC$, so $\angle BAD = \angle CAD = 30^{\circ}$.

Hence, triangle $ABD$ is a right triangle. The hypotenuse has length $AB = 1$ and the side opposite $\angle BAD$ has length $BD = \tfrac{1}{2}$. Hence $\sin 30^{\circ} = \sin \angle BAD = \dfrac{BD}{AB} = \dfrac{1}{2}$.

EDIT: Here I have implicitly used the fact that in an isosceles triangle, the altitude to the base is both the perpendicular bisector of the base and the angle bisector of the vertex opposite of the base. I overlooked this fact as being "trivial", but it does need to be stated. A proof of that fact can be found here. Thanks DonAntonio for pointing that out.

JimmyK4542
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  • See, @Rodrigo? Also here you need some euclidean geometry: why does the perpendicular bisector of any side in an equilateral triangle passes through the opposite vertex? This is false in any other, non-equilateral, triangle (but also true with the perp. bisector of the base in an isoseceles triangle)... – DonAntonio Jun 09 '14 at 21:23
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Perhaps the easiest, most elementary explanation is from basic Euclidean Geometry, with the basic theorem:

"In a right-angled triangle $\;30^\circ-60^\circ-90^\circ\;$, the length of the leg opposite to the $\;30^\circ\;$ angle equals half the hypotenuse's length".

The above gives you at once that, with $\;x:=$ the hypotenuse's length:

$$\;\sin 30^\circ=\frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{\frac x2}x=\frac12$$

DonAntonio
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  • Thanks, but you just repeated the theorem. How did Euclides get there? Why doesn't it give an irrational number, instead of exactly 1/2? – Rodrigo Jun 09 '14 at 21:14
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    @Rodrigo, why $;2+2=4;$ ? Why the median to the base in an isosceles triangles is also its height and the opposite angle's bisector? Why do we humans (usually...) have two legs and not four? Well...because! – DonAntonio Jun 09 '14 at 21:16
  • There is a reason why we have 2 legs, not 4, but it's not as simple and elegant as geometry... :) – Rodrigo Jun 09 '14 at 21:26
  • Well, there's also a reason why $;2+2=4;$ or why the theorem I mention in my answer is true...and you're right: not all reasons are as elegant and nice as in geometry. – DonAntonio Jun 09 '14 at 21:29
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The way $\pi$ was originally defined was in geometric terms: the ratio of a circle's circumference to its radius. If you divide a circle into two equal halves and then divide one of those halves into 6 equal slices, then it stands to reason that the hypotenuse of the triangle is equal to the radius of the original circle, and therefore it follows that $\sin(\frac{\pi}{6}) = \frac{1}{2}$. You can do this without knowing the value of $\pi$ in decimal or sexagesimal or any numeral system whatsoever.

James47
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    in fact the ratio of a circle's circumferente to twice its radius (or to once its diameter) ...and someone asking a question this basic perhaps hasn't yet studied radians. – DonAntonio Jun 09 '14 at 21:27
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Here's another point of view to consider.

I'm thinking that what you're really wondering is why $\sin(\pi/6)$ is such a "nice" number as 1/2.

Instead of looking at it that way, ask yourself, is there some number $x$ such that $\sin(x) = 1/2$? Now if you look at the graph for sin, which shows that it's a smooth function that has a minimum of -1 and a maximum of 1, then it's clear that some such $x$ must exist. Hence, you could say that $\pi/6$ is almost defined to be the number such that $\sin(\pi/6) = 1/2$. Of course, this isn't quite right, but the point is that there's an intricate relationship between the $\sin$ function and $\pi$ that makes the identity $\sin(\pi/6) = 1/2$ true.

oxeimon
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  • Exactly my question... the "niceness" of 1/2, and also of the 30 degrees or pi/6... It could be that sin(29.451364571234...) would be 1/2... but @JimmyK4542 answer explained it well. There is such beauty in such "simple" matters that I still wonder how many mysteries we may draw from there... – Rodrigo Jun 09 '14 at 21:34