Question

Mainly I need help figuring how to get the A and B in the following problem:
Evaluate sin (5pi/12)=
I am curious how this becomes equal to
sin (2pi/3 +(-pi/4) = which would allow me to set and solve the addition formula...

Answer 1

well that is one option... , the question seems to be asking you to get an exact value for

\[\sin(\frac{5\pi}{12})\]

I would use 2 positive values to make life easier

\[\frac{5\pi}{12} = \frac{\pi}{6} + \frac{\pi}{4}\]

and this means that you can get exact values for pi/6 and pi/4

but lookin at the option you posted

\[\frac{5\pi}{12} = \frac{2\pi}{3} - \frac{\pi}{4}\]

this option is ok...but the 1st option is the easier choice...

Answer 2

Thank you very much, however, how did you get pi/6 + pi/4, that is the part that has me confused...

Answer 3

So you can use any two angle in radians, just as long as they add up to 5pi/12 in this case?

Answer 4

Yes, but you really want to choose two angles that are "nice" (most textbooks make you memorize the sin and cos values of pi/6, pi/4, pi/3, and pi/2 for instance)

Answer 5

Thank you so much, that was a huge help... I definitely will try to stick to the the easier ones, but I did just solved a problem that required using radian measure of 3pi/4 + pi/6 to get 11pi/12....

Answer 6

If you're ever having difficult coming up with `the sum of two angles`, you can instead apply the Half-Angle Identity for Sine,\[\Large\rm \sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos(2\theta)}{2}}\]
So we could write \(\Large\rm \dfrac{5\pi}{12}\) as \(\Large\rm\dfrac{(5\pi/6)}{2}\)

And our Half-Angle Formula gives us,\[\Large\rm \sin\left(\frac{5\pi/6}{2}\right)=\sqrt{\frac{1-\cos(5\pi/6)}{2}}\]And then simplify from there.
This is not necessarily an easier approach,
just another option :)

Answer 7

Woops the identity is:
\[\Large\rm \sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos \theta}{2}}\]