If sin(theta)=12/13 when 0<theta<pi/2, what is the value of cos(theta/2)?
To find the value of cos(theta/2), we first need to find the value of theta. We are given that sin(theta) = 12/13 when 0 < theta < pi/2.
From the given information, we can determine that theta is an angle in the first quadrant because sin(theta) is positive. In the first quadrant, both sine and cosine are positive.
Let's solve for theta using the inverse sine function (also known as arcsine):
sin(theta) = 12/13
Taking the inverse sine of both sides:
theta = arcsin(12/13)
Using a calculator, we can find:
theta ≈ 0.934 radians
Now, let's find the value of cos(theta/2):
cos(theta/2) = cos(0.934/2)
Using the half-angle formula for cosine:
cos(theta/2) = ± sqrt((1 + cos(theta)) / 2)
Since theta is in the first quadrant, cos(theta) is positive. Therefore, we can simplify the formula to:
cos(theta/2) = sqrt((1 + cos(theta)) / 2)
Now, substitute the value of theta:
cos(theta/2) = sqrt((1 + cos(0.934)) / 2)
To find the value of cos(0.934), you can use a calculator. After substituting the value, you can simplify further to get the exact value of cos(theta/2).