Use a half angle formula to determine the exact value of sin 7(pi)/12.
well, 7 pi/6 is 180 + 30 degrees
and 7 pi/6 = twice 7 pi/12
we know sin and cos of 7 pi/6
sin 7 pi/6 = -1/2
cos 7 pi/6 = -(1/2)sqrt3
so now you should be able to find the sin of 7 pi/12
To determine the exact value of sin 7π/12, we can make use of the half-angle formula for sine.
The half-angle formula for sine is given by:
sin(θ/2) = ±√[(1 - cosθ) / 2]
In this case, θ = 7π/12.
First, we need to find the value of cos(7π/12). To do this, we can use the double-angle formula for cosine:
cos(2θ) = 2cos²θ - 1
By rearranging this equation, we get:
cosθ = ±√[(1 + cos(2θ)) / 2]
Now, plug in θ = 7π/12 into the equation:
cos(7π/12) = ±√[(1 + cos(2*7π/12)) / 2]
Simplifying further, we have:
cos(7π/12) = ±√[(1 + cos(7π/6)) / 2]
Since cosθ is positive in the second and third quadrants, we choose the positive square root.
Now we know the value of cos(7π/12). Using the half-angle formula for sine, we can calculate sin(7π/12):
sin(7π/12) = ±√[(1 - cos(7π/6)) / 2]
Finally, plug in the value of cos(7π/12):
sin(7π/12) = ±√[(1 - cos(7π/6)) / 2]