Possibility of determining the third side of a right angled triangle using given three parameters:

Question

Suppose there are two arbitrary side lengths of a right angled triangle that are known to us. There are two possible cases here that I can see:

1. Either one of the side lengths given is the length of the hypotenuse.
2. Both the sides lengths given are the lengths of the legs of the right angled triangle

Now, additionally one arbitrary acute angle measure is also known. There are again two possible cases that I can see:

1. The angle lies between the two sides that are known to us (in which case it will then be confirmed that one of the sides given is the hypotenuse).
2. The angle does not lie between the two sides that are known to us (in which case it will then be confirmed that the sides lengths given are the lengths of the legs of the right angled triangle).

Now, I want to know if there is any possibility of determining the third side of the right angled triangle using the conditions given above (two arbitrary sides and one arbitrary acute angle) and without any additional conditions for eliminating the cases given above. If possible, please tell me the formulae/procedure to follow.

Thanks in advance.

Answer 1

If $a$, $b$ are the given lengths of two sides and $\theta$ is the given acute angle, then $a$, $b$ are the legs of the triangle only if $\tan\theta=a/b$ or $\tan\theta=b/a$. Check that to see which case you are dealing with.