Find the exact value by using a half-angle identity. cos 22.5 degrees
use
cos 2A = cos^2A - sin^2A
cos2A = cos^2A - (1 - cos^2A) = 2cos^2A - 1
cos 45° = 2cos^2 22.5° - 1
√2/2 + 1 = 2cos^2 22.5
√2/4 + 1/2 = cos^2 22.5°
cos 22.5° = √[ (√2 + 2)/4 ]
= √(√2 + 2) /2
To find the exact value of cos 22.5 degrees using a half-angle identity, we can use the half-angle identity for cosine:
cos (θ/2) = ±√((1 + cos θ)/2)
Let's substitute θ = 45 degrees into the identity:
cos (45/2) = ±√((1 + cos 45)/2)
First, let's find the value of cos 45 degrees using the unit circle or a trigonometric table. The exact value of cos 45 degrees is 1/√2 or approximately 0.7071.
cos (45/2) = ±√((1 + 1/√2)/2)
Simplifying further, we rationalize the denominator:
cos (45/2) = ±√((2 + √2)/4)
cos (45/2) = ±√(2 + √2)/2
Since 22.5 degrees is in the first quadrant where cos is positive, we take the positive square root:
cos 22.5 degrees = √(2 + √2)/2
Therefore, the exact value of cos 22.5 degrees using a half-angle identity is √(2 + √2)/2.
To find the exact value of cos 22.5 degrees using a half-angle identity, we can use the formula:
cos(θ/2) = ± √[(1 + cosθ) / 2]
In this case, θ = 45 degrees, so the formula becomes:
cos(22.5 degrees) = ± √[(1 + cos 45 degrees) / 2]
Let's evaluate the cosine of 45 degrees first:
cos 45 degrees = (√2 / 2)
Now substituting the value of cos 45 degrees in the half-angle formula:
cos(22.5 degrees) = ± √[(1 + √2 / 2) / 2]
Simplifying further:
cos(22.5 degrees) = ± √[(2 + √2) / 4]
Since 22.5 degrees is in the first quadrant, cos(22.5 degrees) will be positive. Therefore, the exact value of cos 22.5 degrees using a half-angle identity is:
cos(22.5 degrees) = √[(2 + √2) / 4]