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

There are several different ways to answer this question.

I will arbitrarily use

∴

The sine sum identity is:

∴

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

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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) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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