How do you find the exact value of sin(3pi/8) using the DOUBLE ANGLE FORMULAS?
How can I find the exact value of sin(3π/8) using the double angle formulas? I would appreciate a detailed explanation of the steps involved, as I have several similar questions to tackle. Thank you!
2 Answers
cos(2A) = 1 - 2 (sinA)^2
Let A = 3pi/8. Then 2A = 6pi/8 or 3pi/4
cos(3pi/4) is known to be -sqrt(2)/2
So - sqrt(2) / 2 = 1 - 2 (sinA)^2
2 (sinA)^2 = 1 + sqrt(2)/2
(sinA)^2 = (1 + sqrt(2)/2)/2
sq rt of both sides gives the exact value of sin 3pi/8.
Hope that helps!
I don't know how you do it with double-angle formulas, but you can do it with half-angle formulas.
sin(3Ï/8) = sin[(1/2)*(3Ï/4) = â[1 - cos(3Ï/4)]/2
cos(3Ï/4)= cos[(1/2)(3Ï/2)] = -â[1+cos(3Ï/2]/2
cos(3Ï/2) = 0, so cos(3Ï/4) = -â(1/2)
which makes sin(3Ï/8) = â[(1 + â.5)/2] which can be rationalized to (1/2)*â[2+â2]
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