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Monday, 25 March 2013

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

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Example. Check the identities
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Answer. We will check the first one. the second one is left to the reader as an exercise. We have
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Hence
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which implies
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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 tex2html_wrap_inline206 as an expression involving the trigonometric functions with their first power.
Answer. We have
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Hence
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Since tex2html_wrap_inline212 , we get
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or
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Example. Verify the identity
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Answer.We have
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Using the Double-Angle formulas we get
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Putting stuff together we get
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From the Double-Angle formulas, one may generate easily the Half-Angle formulas

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In particular, we have

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Example. Use the Half-Angle formulas to find
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Answer. Set tex2html_wrap_inline232 . Then
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Using the above formulas, we get
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Since tex2html_wrap_inline238 , then tex2html_wrap_inline240 is a positive number. Therefore, we have
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Same arguments lead to
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Example. Check the identities
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Answer. First note that
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which falls from the identity tex2html_wrap_inline250 . So we need to verify only one identity. For example, let us verify that
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using the Half-Angle formulas, we get
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which reduces to
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