Final answer:
To find sin(2u), cos(2u), and tan(2u), we use the sec(u) value to first find sin(u) and cos(u), then apply double-angle formulas. The results are sin(2u) = √3/2, cos(2u) = -1/2, and tan(2u) = -√3.
Step-by-step explanation:
Given that sec(u) = −2 and u is in the interval (π/2, 3π/2), we first need to find the values of sin(u) and cos(u), remembering that sec(u) is the reciprocal of cos(u). Since sec(u) = -2, cos(u) = -1/2. The interval provided indicates that u is in the third quadrant, where sine is also negative, so sin(u) will be negative too. Knowing that the cosine of u is -1/2, we can find the corresponding sine using the Pythagorean identity sin²(u) + cos²(u) = 1, which gives us sin(u) = -√3/2.
Using the double-angle formulas for sine and cosine, we get:
- sin(2u) = 2sin(u)cos(u) = 2*(-√3/2)*(-1/2) = √3/2
- cos(2u) = cos²(u) - sin²(u) = (-1/2)² - (-√3/2)² = 1/4 - 3/4 = -1/2
- tan(2u) = sin(2u) / cos(2u) = (√3/2) / (-1/2) = -√3
We used the angle identities and double-angle formulas to find the exact trigonometric values.