Question: Trying to find cos π/12, if cos π/6 = square root 3 over 2, how to find cos π/12 using DOUBLE angle formula?
This is what I got so far..
cos 2(π/6) = cos (π/6 + π/6)
= (cos π/6)(cos π/6) - (sin π/6)(sin π/6)
= cos^2 π/6 - sin^2 π/6
Is that right? Please and thank you.
you got it backwards
π/12 = (1/2) of π/6
using cos 2A = cos^2 A - sin^2 A = 2cos^2 A - 1
so cos π/6 = 2 cos^2 π/12 - 1
√3/2 + 1 = 2cos^2 π/12
cos^2 π/12 = √3/4 + 1/2 = (√3 + 2)/4
cos π/12 = √(√3+2) /2
check:
by calculator, cos π/12 = .96592...
√(√3+2)/2 = .96592...
My answer is correct.
Of course the answer I gave is not unique
we could have done it this way
cos π/12 = cos 15°
= cos(45-30)°
= cos45cos30 + sin45sin30
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
which when evaluated is also .96592...
but it asked for a solution using the double angle formula
Yes, you're on the right track! To find cos π/12 using the double angle formula, you need to apply the formula to cos 2(π/6), which is correct.
To continue, let's simplify the expression:
cos 2(π/6) = cos^2(π/6) - sin^2(π/6)
Now, to find cos(π/12), we need to use the half angle formula. The half angle formula can be derived from the double angle formula by substituting π/2 for the angle.
Therefore, the half angle formula is:
cos(θ/2) = ±√((1 + cos θ) / 2)
In this case, we can use the formula by substituting π/6 for θ:
cos(π/12) = ±√((1 + cos (π/6)) / 2)
Now, we know that cos(π/6) = √3/2, so we can substitute this value into the formula:
cos(π/12) = ±√((1 + √3/2) / 2)
To simplify further, we need to rationalize the denominator:
cos(π/12) = ±√((2 + √3) / (2*2))
= ±√((2 + √3) / 4)
= ±(√2 + √6) / 2
So, cos(π/12) can be written as ±(√2 + √6) / 2.
It's important to note that the positive or negative sign depends on the quadrant where the angle is located. In this case, π/12 is in the first quadrant, so we take the positive sign: cos(π/12) = (√2 + √6) / 2.
Therefore, cos π/12 is (√2 + √6) / 2.