For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.
For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.
For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.
For the following exercises, prove the identity.
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Numeric
For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.
Technology
For the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.
Extensions
For the following exercises, prove the following sum-to-product formulas.
Start with
Make a substitution and let
and let
so
becomes
Since
and
we can solve for
and
in terms of
x and
y and substitute in for
and get
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For the following exercises, prove the identity.
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