# How do you find the exact functional value sin 75° using the cosine sum or difference identity?

The sine sum identity is:

∴

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

![Unit Circle](

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To find the exact functional value of sin 75° using the cosine sum or difference identity, we can use the identity:

sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)

Here, we can express 75° as the sum of two angles, such that one of the angles has a known sine value. We can choose 75° = 45° + 30°.

Now, we know that sin 45° = cos 45° = 1/√2 and sin 30° = 1/2.

So, applying the identity:

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

Substituting the known values:

sin 75° = (1/√2)(√3/2) + (1/√2)(1/2) = √3/2√2 + 1/2√2

Rationalizing the denominators:

sin 75° = (√3 + 1) / 2√2

Therefore, the exact functional value of sin 75° using the cosine sum or difference identity is (√3 + 1) / 2√2.

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