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How do I use the double or half angle formula to determine the value of the function?
sin(7pi/12)
Can someone show me how to work through this?
Additionally, maybe one with a cosine?
Like, in my homework, I have:
cos(pi/8)
Thanks!
sin(θ/2) = (-1)^|_θ/2π_| · √((1-cos(θ))/2), where |_x_| means "floor of x" ... that is, the greatest integer ≤ x.
For 0 ≤ θ < 2π, 0 ≤ θ/2π < 1, so |_θ/2π_| = 0, so -1^|_θ/2π_| = (-1)^0 = 1.
In this range of angles, the formula simplifies to sin(θ/2) = √((1-cos(θ))/2).
If θ = 7π/6, θ/2 = 7π/12.
sin(7π/12) = sin((7π/6)/2)
= √((1-cos(7π/6))/2).
cos(7π/6) = -√3/2, so
sin(7π/12) = √((1-(-√3/2))/2)
= √((1+√3/2)/2)
= √(1/2 + √3/4).
-------------------------------
cos(θ/2) = (-1)^|_(θ+π)/2π_| · √((1+cos(θ))/2).
If θ = π/4, θ/2 = π/8.
(θ+π)/2π = (π/4 + π)/2π = (5π/4)/2π = 5/8.
5/8 < 1, so |_(θ+π)/2π_| = 0, so -1^|_(θ+π)/2π_| = (-1)^0 = 1.
For this angle, the formula simplifies to cos(θ/2) = √((1+cos(θ))/2).
θ = π/4, so θ/2 = π/8.
cos(π/8) = cos((π/4)/2)
= √((1+cos(π/4))/2).
cos(π/4) = √2/2, so
cos(π/8) = √((1+(√2/2))/2)
= √(1/2 + √2/4).
Half Double Crochet Stitch