suppose sin(x)=5/6, and x is in the first quadrant. Determine cos(x), sin(2x), and cos(2x).

To find cos(x), we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

Given that sin(x) = 5/6, we can substitute this value into the Pythagorean identity:

(5/6)^2 + cos^2(x) = 1

Simplifying,

25/36 + cos^2(x) = 1

cos^2(x) = 1 - 25/36

cos^2(x) = 11/36

Taking the square root of both sides,

cos(x) = ±√(11/36)

Since x is in the first quadrant, cos(x) must be positive. Taking the positive square root,

cos(x) = √(11/36)

To find sin(2x), we can use the double-angle identity: sin(2x) = 2sin(x)cos(x).

Given that sin(x) = 5/6 and cos(x) = √(11/36), we can substitute these values into the double-angle identity:

sin(2x) = 2 * (5/6) * √(11/36)

Simplifying,

sin(2x) = 10√11/36

Finally, to find cos(2x), we can use the double-angle identity: cos(2x) = cos^2(x) - sin^2(x).

Given that sin(x) = 5/6 and cos(x) = √(11/36), we have already found these values earlier.

cos(2x) = cos^2(x) - sin^2(x)

cos(2x) = (11/36) - (25/36)

cos(2x) = -14/36

Simplifying,

cos(2x) = -7/18