2cos^2 22.5-1 (find exact value)
hmmph
It would help to notice that
2cos^2 θ - 1 = cos 2θ
so what you have is just cos 45° = 1/√2
note that 2(2+√2)/4 - 1 = 1+√2/2-1 = √2/2 = 1/√2
cos^2 θ/2 = (1 + cosθ)/2
cos^2 22.5° = (1 + cos 45°)/2
= (1 + √2/2)/2
= (2 + √2)/4
To find the exact value of 2cos^2 22.5 - 1, we can use the double angle identity for cosine.
The double angle identity for cosine states that cos(2θ) = 2cos^2θ - 1.
In our case, θ is equal to 22.5 degrees.
Let's substitute θ = 22.5 degrees into the double angle identity:
cos(2 * 22.5) = 2cos^2(22.5) - 1
cos(45) = 2cos^2(22.5) - 1
Since cos(45) is equal to √2/2, we can rewrite the equation as:
√2/2 = 2cos^2(22.5) - 1
Let's isolate the term 2cos^2(22.5) by adding 1 to both sides:
√2/2 + 1 = 2cos^2(22.5)
Simplifying the left-hand side, we have:
(√2 + 2)/2 = 2cos^2(22.5)
Now, let's divide both sides by 2:
(√2 + 2)/4 = cos^2(22.5)
To find the square root of both sides, we get:
√((√2 + 2)/4) = cos(22.5)
Thus, the exact value of 2cos^2(22.5) - 1 is √((√2 + 2)/4).
To find the exact value of 2cos^2(22.5) - 1, we can use the double-angle formula for cosine. The double-angle formula states that cos(2θ) = 2cos^2(θ) - 1.
Let's use this formula to rewrite the expression: 2cos^2(22.5) - 1 = cos(2(22.5))
Now, we can evaluate cos(2(22.5)) using the cosine function.
To find the exact value of the cosine, we need to convert the angle from degrees to radians, since the trigonometric functions typically work with radians. The conversion formula is π radians = 180 degrees.
Therefore, 22.5 degrees is equal to 22.5 * (π / 180) radians.
Now, let's substitute this value into the expression: cos(2(22.5)) = cos(2 * 22.5 * (π / 180))
Simplifying further, we have cos(2 * 22.5 * (π / 180)) = cos(45π / 180)
Next, we simplify the fraction: 45π / 180 = π / 4
So, cos(45π / 180) = cos(π / 4)
The exact value of cosine π / 4 is 1/√2.
Therefore, 2cos^2(22.5) - 1 = 2 * (1/√2)^2 - 1 = 2 * (1/2) - 1 = 1 - 1 = 0.
Hence, the exact value of 2cos^2(22.5) - 1 is 0.