1 Answers
The half angle formula for cosine states that cos(x/2) = ±sqrt((1 + cos(x))/2), where the sign depends on the quadrant of the angle x/2.
To find the exact value of cos 30 degrees using the half angle formula, we can start by using the identity cos(2x) = 1 - 2sin^2(x) or cos(2x) = 2cos^2(x) - 1.
Let x = 15 degrees. Then we have:
cos(30) = cos(2x) = 2cos^2(x) - 1 = 2cos^2(15) - 1
Now, we can use the half angle formula for cosine with x = 15 degrees:
cos(15) = ±sqrt((1 + cos(30))/2)
Since cos(30) is positive in the first quadrant, we take the positive root:
cos(15) = sqrt((1 + cos(30))/2)
Squaring both sides, we get:
cos^2(15) = (1 + cos(30))/2
Substituting this expression into our earlier equation for cos(30), we get:
cos(30) = 2cos^2(15) - 1 = 2(1 + cos(30))/2 - 1 = cos(30) + 1 - 1 = cos(30)
Therefore, we have shown that cos(30) = cos(30), which is a tautology.
So, the exact value of cos 30 degrees is 1/2.
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