The trigonometric formulas for can be derived using the multiple-angle formula
(1)
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Letting and then gives
(2)
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Factoring out one power of gives
(3)
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Solving the quadratic equation for gives
(4)
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But must be less than
(5)
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so taking the minus sign and simplifying gives
(6)
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Filling in the remainder of the trigonometry functions then gives
(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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and
(13)
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(14)
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(15)
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(16)
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(17)
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(18)
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