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    Find the exact value of (sin22.5° + cos22.5°) without using a calculator.

    Thanx!
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    (Original post by Looontag)
    Find the exact value of (sin22.5° + cos22.5°) without using a calculator.

    Thanx!
    You know the exact values of sin cos and tan of 30,45 and 90.
    The key thing here is to realise 22.5 = (1/2)(45)
    sin22.5+cos22.5=sin(0.5)(45)+cos (0.5)(45)
    You know that cos2A=2(cosA)^2-1 so (cosA)^2=0.5[1+cos2A] and cos2A=1-2(sinA)^2 so (sinA)^2=0.5[1-cos2A]
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    cos45 = cos(2*22.5) = 1 - 2sin²(22.5) = 1/sqrt[2]
    => sin²(22.5) = 1/2 - 1/(2sqrt[2]) = (sqrt[2] - 1)/(2sqrt[2])
    => sin(22.5) = sqrt[(sqrt[2] - 1)/(2sqrt[2])]

    Similarly
    cos45 = 2cos²(2*22.5) - 1 = 1/sqrt[2]
    => cos²(22.5) = 1/(2sqrt[2]) + 1/2 = (sqrt[2] + 1)/(2sqrt[2])
    => cos(22.5) = sqrt[(sqrt[2] + 1)/(2sqrt[2])]

    Hence
    sin(22.5) + cos(22.5) = sqrt[(sqrt[2] - 1)/(2sqrt[2])] + sqrt[(sqrt[2] + 1)/(2sqrt[2])]
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    (Original post by Looontag)
    Find the exact value of (sin22.5° + cos22.5°) without using a calculator.

    Thanx!
    Is this an exam question? If so, and you have your calculator in with you, I wonder how they STOP people from using it?
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    This type of question would tell you if they wanted it in surd form or 'exact values' for or just simply 'not to use a calculator'

    Trust me, they would let you know in one way or another

    Plus, the working would give you the marks... you dont get that with a calculator, although you can double check if you got it right.
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    (Original post by lesser weevil)
    Is this an exam question? If so, and you have your calculator in with you, I wonder how they STOP people from using it?
    They don't need to as it asks for the exact answer and calculators will display it in decimals.

    eg tan(3pi/10)=(1/5)[25+10rt5]^0.5
 
 
 
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