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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:
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:'''S'''ine = '''O'''pposite ÷ '''H'''ypotenuse
:'''C'''osine = '''A'''djacent ÷ '''H'''ypotenuse
:'''T'''angent = '''O'''pposite ÷ '''A'''djacent
 
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."
{{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>
 
Communities exposed to Chinese dialect may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' in Hokkien.
 
==Mnemonic chart==
{{Unreferenced section|date=June 2011}}
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.)
 
[[File:Trigonometric identity mnemonic.png|thumb|Trigonometric identities mnemonic]] Reading across the central 1 in any direction gives reciprocal identities:
 
:<math>
\begin{array}{ccc}
{1 \over \sin A} = \csc A & \text{or} & {1 \over \csc A} = \sin A \\  \\
{1 \over \tan A} = \cot A & \text{or} & {1 \over \cot A} = \tan A \\  \\
{1 \over \sec A} = \cos A & \text{or} & {1 \over \cos A} = \sec A
\end{array}
</math>
 
Reading down any triangle gives the standard identities (starting at the top left and going clockwise):
 
:<math>\sin^2 A + \cos^2 A = 1 \ </math>
 
:<math>1 + \cot^2 A = \csc^2 A \ </math>
 
:<math>\tan^2 A + 1 = \sec^2 A \ </math>
 
Reading a function and dividing the two consecutive clockwise or counter clockwise neighbors gives these identities:
 
(Starting at tan and going clockwise)
 
:<math>\tan A = {\sin A \over \cos A} </math>
 
:<math>\sin A = {\cos A \over \cot A} </math>
 
:<math>\cos A = {\cot A \over \csc A} </math>
 
:<math>\cot A = {\csc A \over \sec A} </math>
 
:<math>\csc A = {\sec A \over \tan A} </math>
 
:<math>\sec A = {\tan A \over \sin A} </math>
 
(Starting at tan and going counter-clockwise)
 
:<math>\tan A = {\sec A \over \csc A} </math>
 
:<math>\sec A = {\csc A \over \cot A} </math>
 
:<math>\csc A = {\cot A \over \cos A} </math>
 
:<math>\cot A = {\cos A \over \sin A} </math>
 
:<math>\cos A = {\sin A \over \tan A} </math>
 
:<math>\sin A = {\tan A \over \sec A} </math>
 
Reading a function and multiplying the two nearest neighbors gives these identities (starting at tan and going clockwise):
 
:<math>\tan A = \sin A \cdot \sec A \ </math>
 
:<math>\sin A = \cos A \cdot \tan A \ </math>
 
:<math>\cos A = \sin A \cdot \cot A \ </math>
 
:<math>\cot A = \cos A \cdot \csc A \ </math>
 
:<math>\csc A = \cot A \cdot \sec A \ </math>
 
:<math>\sec A = \csc A \cdot \tan A \ </math>
 
==References==
{{reflist}}
 
[[Category:Trigonometry]]
[[Category:Mnemonics]]

Latest revision as of 01:24, 26 September 2014

When you are aware what you're undertaking, possessing a splendor schedule are often very pleasurable. These pointers gives you specifics of appropriate elegance strategies.

Make sure you moisturize your face. Even those who have fatty or oily skin area ought to utilize an oil-free of charge moisturizing lotion every single day. Make use of a lotion which contains SPF.

Be sure your foundation complements. Prior to buying a fresh basis, you should check it within your arm. This skin is comparable to your skin layer on your own encounter and will give you an effective illustration of what this makeup will look like on your experience and exactly how it can feel.

Normal water is vital in terms of radiant skin area. When epidermis dehydrates, it lines and wrinkles and dulls. This may be averted by consuming the suggested 8-10 glasses of drinking water every day. If drinking water bores you, use citrus or a tiny cranberry liquid to flavor it. You will observe a spectacular difference in your skin's physical appearance.

To help make shut set up eye seem additional separate, implement your eyes makeup products therefore it is heaver about the outer corners of your view. Use lighting eyeshadow in the inner half of your eyesight and deeper shadow about the outside fifty percent, blending the 2 with each other seamlessly in the center. Then, to end off of the appearance, use your eyeliner and mascara that it is bulkier at the outer spot. This will likely offer the optical illusion that your particular eye are set further more away from each other.

If you have skin area problems, you have to be particularly very careful when picking make up never to worsen the specific situation. Avoid make ups with any gas within them, and sparingly use skin lotions, foundations, vision makeup, and blushes marked gas free. Also try to find non-comedogenic merchandise, which tend not to encourage the closure of pores.

Add lots of fruits and vegetables to the diet program to make your skin layer more wonderful. Eating vegetables and fruit have benefits to every part of your health, not only to the skin. Uncooked food can not only make your skin layer beautiful and vibrant, it might lengthen your daily life by several years.

Make sure that your blush and your lip shade will be in equilibrium. If you use blush, you should ensure that it complements the color you will be employing for the lips. Pinkish ought to be with pinkish, red-colored with red, and so forth. If the shades differ also greatly in one an additional, they will likely conflict very.

Add some Epsom salts in your splendor regimen. Epsom salts may be used to ease your muscle mass and in addition like a laxative. Combine some of the sodium with drinking water along with a tiny lavender to generate a mixture. Utilize this mixture to your tender muscles to assist allievate their pain right away. Your skin layer will look much better by the next day.

You should know seeing that using a proper elegance plan isn't easy. An appropriate routine needs investigation and ought to be implemented frequently. This could be cumbersome but the outcomes are well worth the job and determination. If you utilize the recommendation in this particular piece, you might be soon to be a stunner whoever self assurance is obvious to any or all.

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