Cos(pi/12) solve using half angle formula
cos 2A = 2cos^2 A - 1
cos (π/6) = 2cos^2 (π/12) - 1
√3/2 + 1 = 2cos^2 (π/12)
cos π/12 = ±√(√3/4 + 1/2)
but /12 is in quadrant I
so cos (π/12) = √(√3/4 + 1/2)
or (1/2)√(√3 + 2)
To solve the expression cos(pi/12) using the half angle formula, we need to express pi/12 as a half angle.
The half angle formula for cosine states that cos(x/2) = ± sqrt((1 + cos(x))/2).
To express pi/12 as a half angle, we can consider that 12 is a multiple of 6. Since cos(x) has a periodicity of 2*pi, we can express pi/12 as pi/6 divided by 2.
So, pi/12 = (pi/6) / 2.
Now, we can substitute the value of pi/12 into the formula:
cos(pi/12) = cos( (pi/6) / 2 )
Applying the half angle formula, we get:
cos(pi/12) = ± sqrt((1 + cos(pi/6))/2)
To find the value of cos(pi/6), we can refer to the unit circle. In the unit circle, the cosine of pi/6 is sqrt(3)/2.
Substituting this value into our equation:
cos(pi/12) = ± sqrt((1 + sqrt(3)/2)/2)
To simplify further, we can rationalize the denominator by multiplying both the numerator and denominator by sqrt(2):
cos(pi/12) = ± sqrt(2*(1 + sqrt(3)/2)) / (2 * sqrt(2))
cos(pi/12) = ± sqrt(2 + sqrt(3)) / (2 * sqrt(2))
Therefore, the solution for cos(pi/12) using the half angle formula is ± sqrt(2 + sqrt(3)) / (2 * sqrt(2)).
To solve cos(pi/12) using the half-angle formula, you can start by using the double-angle formula, which states:
cos(2θ) = 2cos^2(θ) - 1
Since the double angle is pi/6 (which is half of pi/12), you can rewrite the formula as:
cos(pi/6) = 2cos^2(pi/12) - 1
Next, let's solve for cos(pi/6). We know that cos(pi/6) is equal to √3/2.
√3/2 = 2cos^2(pi/12) - 1
Rearranging the equation, you have:
2cos^2(pi/12) = √3/2 + 1
Now, divide both sides of the equation by 2:
cos^2(pi/12) = (√3/2 + 1)/2
To solve for cos(pi/12), take the square root of both sides:
cos(pi/12) = √((√3/2 + 1)/2)
Thus, cos(pi/12) is equal to the square root of (√3/2 + 1) divided by 2.