Question

Help me with some Trigonometry please
Picture of the problem will be posted inside

Answer 1

if anyone could help em with this one, id appreciate it, i cant imagine its a very long problem.

Answer 2

Well you need to break apart the 11pi/12 into a sum or difference that will give you an exact value. So if you know values for the sides of a 30-90-60 , 45-90-45 triangle and the identity for sin(A+B) that's helpful.

So let's break apart the 11pi/12 in to 9pi/12 + 2pi/12, since both of these angles are exact values of a 30-90-60 or 45-90-45 triangle.

The next step is write the sin identity with these angles. So we have this:

\[\sin (9\pi/12 + 2\pi/12)= \sin (9\pi/12) \cos(2\pi/12) + \cos (9\pi/12)\sin(2\pi/12)\]
The exact values are as follows:
\[\sin (9\pi/12)=\sqrt{2}/2\]\[\cos (2\pi/12)=\sqrt{3}/2\]\[\cos (9\pi/12)=-\sqrt{2}/2\]\[\sin(2\pi/12)=1/2\] Which gets us:\[\sin(9\pi/12 +2\pi/12) = \sqrt{2}/2 * \sqrt{3}/2 + (-\sqrt{2}/2 * 1/2)\] Which reduces to\[(\sqrt{6}-\sqrt{2})/4\]