Question 750137
I am given a right triangle with angle measures of 15 and 75 degrees.  The side opposite the  15 degree angle is 2 units long.  The question asks me to calculate the exact length of the hypotenuse in radical form.

I doubled the 15 degree angle and created a 30-60-90 triangle so I know the new hypotenuse is twice the new short side, and that the common long side is root three times the short side.  I also have the Pythagorean relations for the two right triangles, so I have a system of equations, but my system resolves to an identity.  I need to introduce some new information.

Thanks,  John


The easiest method is to use the difference of angles formula. 


sin (A – B) = Sin A Cos B – Cos A Sin B


sin (60 - 45) = Sin 60 Cos 45 – Cos 60 Sin 45


sin {{{15^o}}} = Sin 60 Cos 45 – Cos 60 Sin 45


Length of hypotenuse = {{{highlight_green(2(sqrt(6) + sqrt(2)))}}}, or {{{highlight_green(2sqrt(6) + 2sqrt(2))}}} (in simplest radical form)


You can do the check!!


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