Final answer:
The exact values for sin 2u, cos 2u, and tan 2u using the double-angle formulas are -24/25, -7/25, and 24/7 respectively, considering that u is in the fourth quadrant where sin u is negative and cos u is positive.
Step-by-step explanation:
To find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas, we need to use the given information that sin u = −4/5 and that u is in the fourth quadrant (3π/2 < u < 2π). Since u is in the fourth quadrant, cos u is positive.
The double-angle formulas are:
• sin 2u = 2 * sin u * cos u
• cos 2u = cos² u - sin² u = 2 * cos² u - 1 = 1 - 2 * sin² u
• tan 2u = sin 2u / cos 2u
To find cos u, we can use the Pythagorean identity, which states that sin² u + cos² u = 1. Substituting sin u = −4/5 into the equation gives us cos² u = 1 - sin² u = 1 - (−4/5)² = 1 - 16/25 = 9/25. So, cos u = ±3/5. Since u is in the fourth quadrant, we use cos u = 3/5.
Now, we can calculate:
• sin 2u = 2 * (−4/5) * (3/5) = −24/25
• cos 2u = 1 - 2 * (−4/5)² = 1 - 2 * 16/25 = 1 - 32/25 = −7/25
• tan 2u = (−24/25) / (−7/25) = 24/7