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I bet this question has been asked a million times, but I can't find a straight answer. I need to find the length of the hypotenuse in a triangle where I have one side and all the angles.

Example:

Triangle with one side and three angles

Now in the above triangle I have the length of a = 20 and all the angles. How do I - from here - get the length of the hypotenuse (c)?

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    $\begingroup$ according to the formula $c\cdot\cos 30 = a$ $\endgroup$ – W_D Sep 4 '13 at 12:41
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just use Law of sines (https://en.wikipedia.org/wiki/Law_of_sines): it states that

$$\frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}$$

where $\alpha, \beta, \gamma$ are the angles opposited to sides $a, b, c$ respectively. Since $\gamma$ is a right angle, $\sin{\gamma} = 1$, and therefore in your example $c = \frac{a}{\sin{60°}}$.

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  • $\begingroup$ Thanks a lot. Exactly what I needed. Who knew it would be so simple :) $\endgroup$ – Trenskow Sep 4 '13 at 13:05
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    $\begingroup$ This is shooting an ant with a cannon! Just use the right-angle definitions of $\sin$ and $\cos$, as an earlier comment suggested. $a/c = \text{opposite/hypotenuse}=\sin 60^\circ$. $\endgroup$ – Ted Shifrin Sep 4 '13 at 15:02
  • $\begingroup$ if you have a nail to pe put on a wall, everything is a hammer :-) $\endgroup$ – mau Sep 5 '13 at 7:04
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This staple of trigonometry tends to be taught in schools using the mnemonic "SOHCAHTOA":

  • Sine of the angle = Opposite side length / Hypotenuse length.
    • $\sin θ = \frac{Opposite}{Hypotenuse}$
  • Cosine of the angle = Adjacent side length / Hypotenuse length
    • $\cos θ = \frac{Adjacent}{Hypotenuse}$
  • Tangent of the angle = Opposite side length / Adjacent side length
    • $\tan θ = \frac{Opposite}{Adjacent}$

"Adjacent" meaning the side which the given angle joins to the hypotenuse (a if you use the angle 30°), and "Opposite" meaning the side that is not connected to the angle you are using for the calculation (a if you use the angle 60°).

Since both angles are available (even if 60° wasn't stated, you could calculate it as 180 - 90 - 30), you can choose which of the first two to use, flipping the formula to either of these (both are equivalent):

  • Hypotenuse length = Opposite side length / Sine of the angle
    • $Hypotenuse = \frac{Opposite}{\sin θ}$ or in example, $c = \frac{a}{\sin{60°}}$
  • Hypotenuse length = Adjacent side length / Cosine of the angle
    • $Hypotenuse = \frac{Adjacent}{\cos θ}$ or in example, $c = \frac{a}{\cos{30°}}$
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    $\begingroup$ I saw Henrik's comment above saying "in all likelyhood nobody is going to care" about a four-year old question, but this question has over 75,000 views and is on the first page of search results for common searches like "length of hypotenuse from angle", so I think it'd benefit from something that covers the options and jogs the memories of people (like me!) trying to remember this from school... $\endgroup$ – user568458 Jun 15 '18 at 18:40
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Maybe for that problem the short path is seeing the triangle as the half of an equilateral triangle. Therefore $20$ is the height and you know the relationship between the size and the height:

$$h=s\frac{\sqrt{3}}{2} \implies s=\frac{2h}{\sqrt{3}}=\frac{40}{\sqrt{3}}$$

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When it comes to 30 60 90 triangles, the short leg equals half of the hypotenuse, and the long leg equals the short times the square root of three.

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  • $\begingroup$ You're answering a four-and-half-year old question that has an accepted answer, in all likelyhood nobody is going to care. And while correct and usable in this case, this doesn't answer the general question and as you give no argument it's not even a good answer in this case. $\endgroup$ – Henrik May 15 '18 at 20:25

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