Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

(In advanced, I apologize for not knowing how to make fractions)

Here's the problem:

A triangle has side $c = 8$ and angles $A = \pi/4$ and $B = \pi/6$. Find the length of the side opposite $A$.

Here's where I'm at so far:

Since $A$ is 45 degrees and $B$ is 30 degrees, $C$ must be 105 degrees, which is $7\pi/12$ radians.

The law of sines states: $\sin A / a = \sin B / b = \sin C / c$

So, we can say $\sin(\pi/4) / a = \sin(7\pi/12) / 8$.

Here's where I'm stuck:

This question may seem so novice, but assuming that I'm correct up until this point (please indicate if I'm not), how do I solve this now? The $\sin(7\pi/12)$ is not a value determined from either a 30-60-90 triangle or a 45-45-90 triangle, so I'm not sure what to do with it.

Any and all help is greatly appreciated!


Thanks to Dylan's comment, I got $16\sqrt2 / (\sqrt2 + \sqrt6)$. Silly me though, I forget how to rationalize the denominator... any help in this area is also appreciated.

Another edit: The furthest I got this rationalized so far is $16 / (1 + \sqrt3$) by first multiplying by $sqrt2$ / $sqrt2$. Then, I got $8\sqrt3 - 8$ by multiplying by $1 - \sqrt3$ / $1 - \sqrt3$. Is this correct?

share|cite|improve this question
you have a typo, C must be 105 degrees is $\frac{7\pi{}}{12}$ radians. – user12205 Sep 14 '11 at 1:43
$7\pi/12 = \pi/4 + \pi/3$, so you could use the addition formula for $\sin$, which would involve computations you seem to be comfortable with. – Dylan Moreland Sep 14 '11 at 1:45
by the way, you can make fractions using \frac{numerator}{denominator} – user12205 Sep 14 '11 at 1:49
@Jer: I tried that, but got swept away in a sea of edits! – The Chaz 2.0 Sep 14 '11 at 1:50
@Mike It doesn't appear that you're having trouble with that at all! – Dylan Moreland Sep 14 '11 at 2:08
up vote 1 down vote accepted

$$ \sin 105^\circ = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ. $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.