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We have just studied the law of sines. I am trying to find the ratio $a:b:c$ of the sides of a triangle with angles $\alpha=120^\circ$ and $\beta=30^\circ.$

We can calculate the third angle of the triangle: $\gamma=180^\circ-\alpha-\beta=30^\circ.$ So the triangle is isosceles. The law of sines gives us $$a:b:c=\sin\alpha:\sin\beta:\sin\gamma=$$ $$=\sin120^\circ:\sin30^\circ:\sin30^\circ=\dfrac{\sqrt3}{2}:\dfrac12:\dfrac12=\left(\dfrac{\sqrt3}{2}\cdot2\right):\dfrac{1}{2}=\text{...}=2\sqrt3.$$ That can't be it! Where am I wrong?

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  • $\begingroup$ I don't understand. Have I made a mistake somewhere? $\endgroup$
    – Katherine
    Jan 6 at 18:49
  • $\begingroup$ We don't understand. You found $\sqrt{3}:1:1$. What're you writing after it, in your post? $\endgroup$
    – cosmo5
    Jan 6 at 18:50
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    $\begingroup$ $\sqrt{3}/2:1/2:1/2 = \sqrt{3}:1:1$ on doubling each. $\endgroup$
    – cosmo5
    Jan 6 at 18:51
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    $\begingroup$ @nicoledobreva: You seem to be taking the ratio indicator "$:$" as a division operator. This arguably isn't "wrong" in a two-term ratio (we often conflate $a:b$ and $\frac{a}{b}$), but it doesn't make sense in a multi-term one. So, once you have $a:b:c$, all you can do is simplify by multiplying- or dividing-through by a common (non-zero) value. $\endgroup$
    – Blue
    Jan 6 at 19:00
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    $\begingroup$ @nicoledobreva: "By "multiplying- or dividing-through by a common value" you mean I can multiply (or divide) each of the terms by a fixed number, right?" Right. Like @ cosmo5 did in getting from your answer to $\sqrt3:1:1$ by "doubling each". $\endgroup$
    – Blue
    Jan 6 at 19:07
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You are correct; $a:b:c=\sqrt3:1:1$.

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  • $\begingroup$ But I got that $a:b:c=2\sqrt3$. How do we conclude $a:b:c=\sqrt3:1:1$? $\endgroup$
    – Katherine
    Jan 6 at 18:50
  • $\begingroup$ How are you turning a ratio of $3$ numbers into a single number? $\endgroup$ Jan 6 at 18:51
  • $\begingroup$ @nicoledobreva You have written something that is not even wrong. $\endgroup$ Jan 6 at 18:51
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    $\begingroup$ @nicoledobreva $\frac{\sqrt{3}}{2}:\frac{1}{2}:\frac{1}{2}=\frac{\sqrt{3}}{2}\times2:\frac{1}{2}\times2:\frac{1}{2}\times2=\sqrt{3}:1:1$ $\endgroup$
    – basilisk
    Jan 6 at 18:54
  • $\begingroup$ Thank you! I appreciate your help. $\endgroup$
    – Katherine
    Jan 6 at 18:59

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