find the exact value:
cos^-1 (square root of 2 divided by 2)
draw a 45 45 90 triangle with equal legs = 1
hypotenuse^2 = 1^2+1^2
h^2 = 2
h = sqrt 2
then cos 45 = 1/sqrt2 = sqrt 2/2
To find the exact value of cos^(-1)(√2/2), we need to use the inverse cosine function, also known as arccosine. We know that arccosine gives us the angle whose cosine is equal to a given value.
The given value in this case is (√2/2). Let's denote it as x, where x = (√2/2):
x = (√2/2)
To find the exact value of arccosine (√2/2), we need to determine the angle whose cosine is equal to (√2/2) in the interval [0, π] (radians) or [0, 180°] (degrees).
By finding the reference angle, we can determine the quadrant in which the angle lies and use it to find the exact value. For cosine values, we commonly use the unit circle to determine the reference angle.
In this case, (√2/2) is a common value obtained from the unit circle at 45° or π/4 rad. Since cosine is positive in the first and fourth quadrants, we can conclude that the angle whose cosine is (√2/2) is 45° or π/4 rad.
Therefore, the exact value of cos^(-1)(√2/2) is 45° or π/4 rad.