Show that Cos 22.5=√2+√2/2
22.5° = 45° / 2
cos 45° = √ 2 / 2
cos ( θ / 2 ) = ± √ [ ( 1 + cos θ ) / 2 ]
45° is located in first quadrant where all trigonometric functions are positive so:
cos 22.5° = cos ( 45° / 2 ) = √ [ ( 1 + cos 45° ) / 2 ] =
√ [ ( 1 + √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 + √ 2 / 2 ) / 2 ] =
√ [ ( 2 + √ 2 ) / 2 / 2 ] = √ [ ( 2 + √ 2 ) / 4 ] =
√ ( 2 + √ 2 ) / √ 4 = √ ( 2 + √ 2 ) / 2
To show that cos 22.5° equals √2 + √2/2, we can make use of the half-angle formula for cosine.
The half-angle formula for cosine states that cos(θ/2) = ±√((1 + cosθ) / 2), where θ is the original angle.
In this case, we have θ = 45°, so we want to find cos(45°/2) = cos(22.5°).
Plug the value of θ into the half-angle formula:
cos(22.5°) = ±√((1 + cos(45°)) / 2)
Since cos(45°) = √2/2, we can substitute it into the equation:
cos(22.5°) = ±√((1 + √2/2) / 2)
Next, simplify the expression inside the square root:
cos(22.5°) = ±√((2 + √2) / 4)
Multiply both the numerator and denominator by 2 to simplify further:
cos(22.5°) = ±√(2 + √2) / √4)
The square root of 4 simplifies to 2:
cos(22.5°) = ±√(2 + √2) / 2
Now, if we consider the positive square root:
cos(22.5°) = √(2 + √2) / 2
This expression matches the desired result of √2 + √2/2.
Therefore, we have shown that cos 22.5° equals √2 + √2/2.