if Ø is an angle in the first quadrant and cosØ= 1/2, find the exact value of cos2Ø
This is almost the same kind of question that I answered for you here :
http://www.jiskha.com/display.cgi?id=1456693077
Did you look at it, and learn from it?
I will give you one hint for this one:
cos (2Ø) = cos^2 Ø - sin^2 Ø
let me know how you made out.
oh ok i think i understand how to do them know...thank you
Tell me what you got, then we'll both be happy
To find the exact value of cos(2Ø), we'll need to use a trigonometric identity.
First, let's find the value of sin(Ø) using the Pythagorean identity: sin²(Ø) + cos²(Ø) = 1.
Since Ø is in the first quadrant, sin(Ø) is positive.
Given that cos(Ø) = 1/2, we can find sin(Ø):
sin²(Ø) = 1 - cos²(Ø) = 1 - (1/2)² = 1 - 1/4 = 3/4.
Taking the square root, we find sin(Ø) = √(3/4) = √3 / 2.
Now, let's use the double-angle identity for cosine to find cos(2Ø):
cos(2Ø) = cos²(Ø) - sin²(Ø).
Substituting the values we found earlier:
cos(2Ø) = (1/2)² - (√3 / 2)² = 1/4 - 3/4 = -2/4 = -1/2.
So, the exact value of cos(2Ø) when cos(Ø) = 1/2 is -1/2.