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# And another trig question :-( watch

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

Thanx!
2. (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]
3. 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])]
4. (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?
5. 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.
6. (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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