Find sin 2x, cos 2x, and tan 2x if cos x = 3 / (sqrt (13)) and x terminates in quadrant I.

sin (2x) = 12/13

cos (2x) = 5/13

tan (2x) = 12/5

Step-by-step explanation:

cos x = 3/√13, x is in the first quadrant.

Use Pythagorean identity to find sin x.

sin²x + cos²x = 1

sin²x + (3/√13) ² = 1

sin²x + 9/13 = 1

sin²x = 4/13

sin x = ±2/√13

Since x is in the first quadrant, sin x = 2/√13.

Use double angle formulas:

sin (2x) = 2 sin x cos x

sin (2x) = 2 (2/√13) (3/√13)

sin (2x) = 12/13

cos (2x) = cos²x - sin²x

cos (2x) = (3/√13) ² - (2/√13) ²

cos (2x) = 9/13 - 4/13

cos (2x) = 5/13

tan (2x) = sin (2x) / cos (2x)

tan (2x) = (12/13) / (5/13)

tan (2x) = 12/5