Use the half angle formula to find
cos(pi/12)
Hint #1 : pi/12 = 1/2 of pi/6, and you should be familiar with the trig ratios of pi/6 or 30º
Hint #2 : cos 2A = 2cos^2 A - 1
To find cos(pi/12) using the half angle formula, we can start by first recalling the half angle formula for cosine:
cos(theta/2) = ± sqrt( (1 + cos(theta)) / 2 )
In this case, we want to find cos(pi/12), so we substitute theta = pi/6 into the formula:
cos(pi/12/2) = ± sqrt( (1 + cos(pi/6)) / 2 )
Next, we need to calculate cos(pi/6). To do so, we can refer to the special triangle associated with the 30-60-90 degrees triangle.
In a 30-60-90 degrees triangle, the sides are related by the ratio:
opposite of 30 degrees angle : opposite of 60 degrees angle : hypotenuse = 1 : sqrt(3) : 2
Since cos(theta) = adjacent/hypotenuse in a right triangle, in our case, cos(pi/6) = adjacent/2. Therefore, the adjacent side length is 2 * cos(pi/6).
To find cos(pi/6), we can use the fact that cos(pi/6) = sqrt(3)/2.
So, cos(pi/6) = sqrt(3)/2
Plugging this back into the half angle formula:
cos(pi/12/2) = ± sqrt( (1 + sqrt(3)/2) / 2 )
Simplifying further:
cos(pi/12/2) = ± sqrt( (2 + sqrt(3)) / 4 )
Now, we are left with two possible values for cos(pi/12/2) - one positive and one negative, indicated by the ± sign. To determine which one is correct, we need to consider the given angle.
Since pi/12 is in the first quadrant (between 0 and pi/2), the cosine function is positive. Therefore, we can take the positive value:
cos(pi/12/2) = sqrt( (2 + sqrt(3)) / 4 )
Now, we can simplify:
cos(pi/12) = [sqrt(2 + sqrt(3))] / [sqrt(4)]
Finally, simplifying again:
cos(pi/12) = sqrt(2 + sqrt(3)) / 2
So, the value of cos(pi/12), using the half angle formula, is sqrt(2 + sqrt(3)) / 2.