Use the product-to-sum identities to rewrite the following expression as a sum or difference.3sin(80°)cos(65°)
Question
Use the product-to-sum identities to rewrite the following expression as a sum or difference.3sin(80°)cos(65°)
Solution
The product-to-sum identities in trigonometry are:
sin(a)cos(b) = 1/2[sin(a+b) + sin(a-b)] cos(a)sin(b) = 1/2[sin(a+b) - sin(a-b)] cos(a)cos(b) = 1/2[cos(a+b) + cos(a-b)] sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)]
The given expression is 3sin(80°)cos(65°).
We can use the first identity to rewrite this expression:
3sin(80°)cos(65°) = 3/2[sin(80°+65°) + sin(80°-65°)]
Simplify the expression inside the brackets:
= 3/2[sin(145°) + sin(15°)]
So, the expression 3sin(80°)cos(65°) can be rewritten as a sum using the product-to-sum identities as 3/2[sin(145°) + sin(15°)].
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