help please. find the exact value of each. 1.) cot 25 π / 12 2.) csc 22.5 °?

solutions are very much appreciated as much as answers :)

THANK YOU :)

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  • 9 years ago
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    Question - 1:

    i) cot (25π/12) = cot{(24π + π) / 12} = cot(2π + π/12) = cot(π/12) = cot(15°)

    ii) By trigonometric multiple angle identities,

    1 + cos(2θ) = 2cos²θ and sin(2θ) = 2sin(θ)*cos(θ)

    ==> {1 + cos(2θ)}/{sin(2θ)} = 2cos²θ/ 2sin(θ)*cos(θ) = cot(θ)

    iii) Hence of the above, cot(15°) = {1 + cos(30°)}/sin(30°) = (1 + √3/2)/(1/2) = 2 + √3

    Thus value of cot(25π/12) = 2 + √3 = 3.732 (Nearly)

    Question - 2:

    i) As of the above application, cot(22.5°) = {1 + cos(45°)}/sin(45°)

    = (1 + 1/√2)/(1/√2) = √2 + 1

    ii) Squaring. cot²22.5° = 3 + 2√2

    iii) Adding 1 both sides, cot²22.5° + 1 = 4 + 2√2

    ==> csc²22.5° = 4 + 2√2

    Hence, csc(22.5°) = √(4 + 2√2) = 2.6131 (Nearly)

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