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| In [[trigonometry]], it is common to use [[mnemonic]]s to help remember [[trigonometric identities]] and the relationships between the various [[trigonometric functions]]. For example, the ''sine'', ''cosine'', and ''tangent'' ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English:
| | I am 22 years old and my name is Stephaine Dwight. I life in Vesoul (France).<br><br>Here is my site :: [http://y5y.pl darmowe subdomeny] |
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| :'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse
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| :'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse
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| :'''T'''angent = '''O'''pposite ÷ '''A'''djacent
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| One way to remember the letters is to sound them out phonetically (i.e. "SOH-CAH-TO-A").<ref>{{MathWorld|title=SOHCAHTOA|urlname=SOHCAHTOA}}</ref> Another method is to expand the letters into a sentence, such as "'''S'''ome '''O'''ld '''H'''ippie '''C'''aught '''A'''nother '''H'''ippie '''T'''rippin' '''O'''n '''A'''cid".<ref> A sentence that is more appropriate for high school is, "Some old horse came a'hopping through our alley."
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| {{cite book |title=Memory: A Very Short Introduction|first=Jonathan K.|last=Foster|publisher=Oxford|year=2008|isbn=0-19-280675-0|page=128}}</ref>
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| Communities exposed to Chinese dialect may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' in Hokkien.
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| ==Mnemonic chart==
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| {{Unreferenced section|date=June 2011}}
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| Another mnemonic permits all of the basic identities to be read off quickly. Although the word part of the mnemonic used to build the chart does not hold in English, the chart itself is fairly easy to reconstruct with a little thought. (Functions without "co" appear on the left, co-functions on the right, a 1 goes in the middle, triangles point down, and the entire drawing looks like a radiation symbol.)
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| [[File:Trigonometric identity mnemonic.png|thumb|Trigonometric identities mnemonic]] Reading across the central 1 in any direction gives reciprocal identities:
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| :<math> | |
| \begin{array}{ccc}
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| {1 \over \sin A} = \csc A & \text{or} & {1 \over \csc A} = \sin A \\ \\
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| {1 \over \tan A} = \cot A & \text{or} & {1 \over \cot A} = \tan A \\ \\
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| {1 \over \sec A} = \cos A & \text{or} & {1 \over \cos A} = \sec A
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| \end{array}
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| </math>
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| Reading down any triangle gives the standard identities (starting at the top left and going clockwise):
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| :<math>\sin^2 A + \cos^2 A = 1 \ </math>
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| :<math>1 + \cot^2 A = \csc^2 A \ </math>
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| :<math>\tan^2 A + 1 = \sec^2 A \ </math>
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| Reading a function and dividing the two consecutive clockwise or counter clockwise neighbors gives these identities:
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| (Starting at tan and going clockwise)
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| :<math>\tan A = {\sin A \over \cos A} </math>
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| :<math>\sin A = {\cos A \over \cot A} </math>
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| :<math>\cos A = {\cot A \over \csc A} </math>
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| :<math>\cot A = {\csc A \over \sec A} </math>
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| :<math>\csc A = {\sec A \over \tan A} </math>
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| :<math>\sec A = {\tan A \over \sin A} </math>
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| (Starting at tan and going counter-clockwise)
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| :<math>\tan A = {\sec A \over \csc A} </math>
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| :<math>\sec A = {\csc A \over \cot A} </math>
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| :<math>\csc A = {\cot A \over \cos A} </math>
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| :<math>\cot A = {\cos A \over \sin A} </math>
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| :<math>\cos A = {\sin A \over \tan A} </math>
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| :<math>\sin A = {\tan A \over \sec A} </math>
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| Reading a function and multiplying the two nearest neighbors gives these identities (starting at tan and going clockwise):
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| :<math>\tan A = \sin A \cdot \sec A \ </math>
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| :<math>\sin A = \cos A \cdot \tan A \ </math>
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| :<math>\cos A = \sin A \cdot \cot A \ </math>
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| :<math>\cot A = \cos A \cdot \csc A \ </math>
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| :<math>\csc A = \cot A \cdot \sec A \ </math>
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| :<math>\sec A = \csc A \cdot \tan A \ </math>
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| ==References==
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| {{reflist}}
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| [[Category:Trigonometry]]
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| [[Category:Mnemonics]]
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I am 22 years old and my name is Stephaine Dwight. I life in Vesoul (France).
Here is my site :: darmowe subdomeny