use a half angle formula to find the exact value of cos 75 degree
To find the exact value of cos 75 degrees using a half angle formula, we will start by using the double angle formula for cosine:
cos 2θ = 2 * cos^2 θ - 1
Let's solve for cos θ first. We know that θ = 2 * (75/2) = 75 degrees.
cos θ = cos (75 degrees / 2) = √[(1 + cos 75 degrees)/2]
Now let's substitute cos θ back into the double angle formula equation:
cos 2θ = 2 * cos^2 θ - 1
cos 150 degrees = 2 * (√[(1 + cos 75 degrees)/2])^2 - 1
cos 150 degrees = 2 * [(1 + cos 75 degrees)/2] - 1
cos 150 degrees = 1 + cos 75 degrees - 1
cos 150 degrees = cos 75 degrees
So, cos 75 degrees = cos 150 degrees.
By using a half angle formula, we can conclude that the exact value of cos 75 degrees is the same as cos 150 degrees. Now we need to find the exact value of cos 150 degrees.
To find the exact value of cos 150 degrees, we need to use the unit circle or reference angles.
On the unit circle, we know that the cosine value at 150 degrees is negative since it lies in the second quadrant. Also, the cosine value at 30 degrees is √3/2.
cos 30 degrees = √3/2
Since we are dealing with cos 150 degrees, which is in the second quadrant, we need to use the negative value:
cos 150 degrees = -√3/2
Therefore, the exact value of cos 75 degrees using the half angle formula is -√3/2.
To find the exact value of cos 75 degrees using a half angle formula, we will use the formula for cos(θ/2):
cos(θ/2) = ± sqrt((1 + cosθ) / 2)
First, we'll find the value of cos 150 degrees using the given half angle formula:
cos(150/2) = ± sqrt((1 + cos150) / 2)
cos(150/2) = ± sqrt((1 + (-sqrt(3)/2)) / 2)
cos(150/2) = ± sqrt((1 - sqrt(3)/2) / 2)
cos(150/2) = ± sqrt((2 - sqrt(3)) / 4)
Since we need to find the value of cos 75 degrees, which is half of 150 degrees, we can substitute 75 instead of 150:
cos(75/2) = ± sqrt((2 - sqrt(3)) / 4)
Now, let's simplify the expression further:
cos(75/2) = ± sqrt(2 - sqrt(3)) / sqrt(4)
cos(75/2) = ± sqrt(2 - sqrt(3)) / 2
Thus, the exact value of cos 75 degrees using the half angle formula is ± sqrt(2 - sqrt(3)) / 2.
start with
cos 2A = 2cos^2 A - 1
cos150° = 2 cos^2 75 - 1
-√3/2 + 1 = 2cos^2 75°
cos^2 75° = (1 - √3/2)/2
cos 75° = √[(1 - √3/2)/2]
An easier ways would have been
cos75 = cos(45+30)
= cos45cos30 - sin45sin30
= (1/√2)(√3/2) - (1/√2)(1/2)
= (√3 - 1)/(2√2)
Use your calculator to show that both produce cos75