How do you find the exact functional value sin 11/12 pi using the cosine sum or difference identity?

Question

Anna Andrews · Accepted Answer

#color(red)(sin((11π)/12) = (sqrt3-1)/(2sqrt2))#

There are several different ways to answer this question.

I will arbitrarily use

#(11π)/12 = (8π)/12 + (3π)/12 = (2π)/3+π/4#

∴ #sin((11π)/12)= sin((2π)/3+π/4)#

The sine sum identity is:

#sin(A+B) = sinAcosB+cosAsinB#

∴ #sin((11π)/12) = sin((2π)/3)cos(π/4) + cos((2π)/3)sin(π/4)#

We can use the unit circle to work out the values.

Carter Anderson · Answer

undefined To find the exact functional value of $ \sin \left(\frac{11}{12} \pi \right) $ using the cosine sum or difference identity, you can utilize the identity $ \sin(\theta) = \cos \left(\frac{\pi}{2} - \theta \right) $.

First, rewrite $ \frac{11}{12} \pi $ as $ \frac{\pi}{2} - \left(\frac{\pi}{2} - \frac{11}{12} \pi \right) $.

Then, use the cosine sum or difference identity:
$$ \cos(A - B) = \cos A \cos B + \sin A \sin B $$

Here, $ A = \frac{\pi}{2} $ and $ B = \frac{\pi}{2} - \frac{11}{12} \pi $, so:
$$ \cos \left(\frac{\pi}{2} - \frac{\pi}{2} + \frac{11}{12} \pi \right) = \cos \left(\frac{11}{12} \pi \right) $$

$$ \cos \left(\frac{\pi}{2} - \frac{\pi}{2} + \frac{11}{12} \pi \right) = \cos \left(\frac{\pi}{2} \right) \cos \left(\frac{11}{12} \pi \right) + \sin \left(\frac{\pi}{2} \right) \sin \left(\frac{11}{12} \pi \right) $$

$$ \cos \left(\frac{\pi}{2} - \frac{\pi}{2} + \frac{11}{12} \pi \right) = 0 \times \cos \left(\frac{11}{12} \pi \right) + 1 \times \sin \left(\frac{11}{12} \pi \right) $$

$$ \cos \left(\frac{11}{12} \pi \right) = \sin \left(\frac{11}{12} \pi \right) $$

Therefore, $ \sin \left(\frac{11}{12} \pi \right) = \cos \left(\frac{\pi}{2} - \frac{11}{12} \pi \right) $.