Find sin(2x), cos(2x), and tan(2x) from the given information.tan(x) = − 125,   x in Quadrant II

To find sin(2x), cos(2x), and tan(2x) given that tan(x) = -125 and x is in Quadrant II, we first need to find sin(x) and cos(x).

Step 1: Find sin(x) and cos(x) Since tan(x) = sin(x)/cos(x), we know that sin(x) and cos(x) must be in the ratio -125:1. However, since x is in Quadrant II, cos(x) is negative and sin(x) is positive. Therefore, sin(x) = 125/√(1² + (-125)²) = 125/√15626 = 125/125 = 1 and cos(x) = -1/√15626 = -1/125.

Step 2: Find sin(2x) and cos(2x) We can use the double angle formulas sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x). Plugging in the values we found for sin(x) and cos(x), we get sin(2x) = 21(-1/125) = -2/125 and cos(2x) = (-1/125)² - 1² = 1/15625 - 1 = -15624/15625.

Step 3: Find tan(2x) Finally, we can find tan(2x) = sin(2x)/cos(2x) = (-2/125) / (-15624/15625) = 15625/1562400 = 0.01.

So, sin(2x) = -2/125, cos(2x) = -15624/15625, and tan(2x) = 0.01.