If x is an angle in the first quadrant and cscx = 3 find the exact value of cos 2x
if cscx = 3, then sinx = 1/3
by Pythagoras, cosx = √8/3
cos 2x = cos^2x - sin^2x = 8/9 - 1/9 = 7/9
To find the exact value of cos 2x, we can use the identity cos 2x = 1 - 2sin²x.
Given that csc x = 3, we can find the value of sin x by taking the reciprocal: sin x = 1/3.
Since x is an angle in the first quadrant, both sine and cosine will be positive.
Now, we can find the value of cos x using the Pythagorean identity: cos²x = 1 - sin²x.
Substituting sin²x = (1/3)² = 1/9 into the equation, we get:
cos²x = 1 - 1/9
cos²x = 8/9
Taking the square root of both sides (remembering that cos x is positive), we find:
cos x = √(8/9) = (√8)/(√9) = √8/3
Now, we can find the exact value of cos 2x using the identity cos 2x = 1 - 2sin²x:
cos 2x = 1 - 2(sin x)²
cos 2x = 1 - 2(1/3)²
cos 2x = 1 - 2/9
cos 2x = (9 - 2)/9
cos 2x = 7/9
Therefore, the exact value of cos 2x is 7/9.