Question

what is the value of 2sin π/32

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Answer 1

Answer:

The half angle formulas are:

sin(B/2) = ± √([1 − cos B] / 2)

cos(B/2) = ± √([1 + cos B] / 2)

But π/32 is in the first quadrant, so we can just ignore the minus.

Use it first to try to find sin(π/32):

sin(π/32) = √([1 − cos π/16] / 2)

But you don't know cos(π/16), so use the cos half-angle formula to find that:

cos(π/16) = √([1 + cos π/8] / 2)

But again, you don't know cos π/8, so AGAIN use the half-angle formula to find that:

cos(π/8) = √([1 + cos π/4] / 2)

Finally! You know that cos π/4 = √2/2, so:

cos(π/8) = √([1 + √2/2] / 2)

Now you can go back and plug that into the previous one:

cos(π/16) = √([1 + √([1 + √2/2] / 2)] / 2)

And then you can plug THAT into the original one:

sin(π/32) = √([1 − √([1 + √([1 + √2/2] / 2)] / 2)] / 2)

Then double it to get your answer:

2sin(π/32) = 2√([1 − √([1 + √([1 + √2/2] / 2)] / 2)] / 2)

2√([1 − √([1 + √([(2 + √2)/2] / 2)] / 2)] / 2)

2√([1 − √([1 + √((2 + √2)/4)] / 2)] / 2)

2√([1 − √([1 + √(2 + √2)/2] / 2)] / 2)

2√([1 − √([2 + √(2 + √2))/2] / 2)] / 2)

2√([1 − √((2 + √(2 + √2))/4)] / 2)

2√([1 − √(2 + √(2 + √2))/2] / 2)

2√([(2 − √(2 + √(2 + √2)))/2] / 2)

2√((2 − √(2 + √(2 + √2)))/4)

2√(2 − √(2 + √(2 + √2)))/2

√(2 − √(2 + √(2 + √2)))

√(2 − √(2 + √(2 + √2))) ≈ 0.1960

2sin π/32 ≈ 0.1960

Answer 2

Step-by-step explanation:

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