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					Double-Angle and Half-Angle Formulas


Double-Angle and Half-Angle formulas are very useful. For example, rational functions
of sine and cosine wil be very hard to integrate without these formulas. They are as
follow




Example. Check the identities




Answer. We will check the first one. the second one is left to the reader as an exercise.
We have



Hence



which implies




Many functions involving powers of sine and cosine are hard to integrate. The use of
Double-Angle formulas help reduce the degree of difficulty.


Example. Write            as an expression involving the trigonometric functions with
their first power.
Answer. We have




Hence




Since                            , we get




or




Example. Verify the identity



Answer.We have




Using the Double-Angle formulas we get




Putting stuff together we get




From the Double-Angle formulas, one may generate easily the Half-Angle formulas
In particular, we have




Example. Use the Half-Angle formulas to find




Answer. Set          . Then




Using the above formulas, we get




Since               , then         is a positive number. Therefore, we have
Same arguments lead to




Example. Check the identities




Answer. First note that




which falls from the identity               . So we need to verify only one
identity. For example, let us verify that




using the Half-Angle formulas, we get




which reduces to

				
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posted:8/19/2011
language:English
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