To find the exact values of $\sin(2u)$, $\cos(2u)$, and $\tan(2u)$, we will use the double angle formulas for trigonometric functions. The given condition is:

• $\sin(u) = -\frac{4}{5}$
• $\frac{3\pi}{2} < u < 2\pi$, indicating that $u$ is in the fourth quadrant.

Step 1: Find $\cos(u)$

We know the Pythagorean identity:

$\sin^2(u) + \cos^2(u) = 1$

Substituting $\sin(u) = -\frac{4}{5}$:

$\left(-\frac{4}{5}\right)^2 + \cos^2(u) = 1$ $\frac{16}{25} + \cos^2(u) = 1$ $\cos^2(u) = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}$ $\cos(u) = \pm \frac{3}{5}$

Since $u$ is in the fourth quadrant, where $\cos(u)$ is positive, we have:

$\cos(u) = \frac{3}{5}$

Step 2: Use the Double Angle Formulas

The double angle formulas are:

$\sin(2u) = 2\sin(u)\cos(u)$ $\cos(2u) = \cos^2(u) - \sin^2(u)$ $\tan(2u) = \frac{\sin(2u)}{\cos(2u)}$

Step 3: Calculate $\sin(2u)$, $\cos(2u)$, and $\tan(2u)$

1. Find $\sin(2u)$:

$\sin(2u) = 2\sin(u)\cos(u) = 2\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right)$ $\sin(2u) = -\frac{24}{25}$

2. Find $\cos(2u)$:

$\cos(2u) = \cos^2(u) - \sin^2(u) = \left(\frac{3}{5}\right)^2 - \left(-\frac{4}{5}\right)^2$ $\cos(2u) = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25}$

3. Find $\tan(2u)$:

$\tan(2u) = \frac{\sin(2u)}{\cos(2u)} = \frac{-\frac{24}{25}}{-\frac{7}{25}} = \frac{24}{7}$

Final Answers:

$\sin(2u) = -\frac{24}{25}, \quad \cos(2u) = -\frac{7}{25}, \quad \tan(2u) = \frac{24}{7}$

Would you like any further details or have additional questions about the steps? Here are some related questions for further practice:

1. How do you derive the double angle formulas for trigonometric functions?
2. What is the significance of the signs of trigonometric functions in different quadrants?
3. How would you solve a similar problem if the given information was for $\cos(u)$ instead of $\sin(u)$?
4. How can the double angle formulas be used to simplify expressions involving trigonometric functions?
5. Can you apply these formulas to find trigonometric values for half-angles (using half-angle identities)?

Tip: Always consider the quadrant when determining the signs of trigonometric functions—this can help you decide whether to use positive or negative values for $\sin(u)$, $\cos(u)$, and $\tan(u)$.