Let's analyze the given figure and solve for $\sin(2u)$, $\cos(2u)$, and $\tan(2u)$.

The figure shows a vector $\mathbf{v}$ with its terminal point at $(1, 2)$.

First, we need to determine the angle $u$. To find $u$, we will use the coordinates of the vector $(1, 2)$. The angle $u$ is the angle that the vector makes with the positive $x$-axis.

The tangent of angle $u$ can be found using: $\tan(u) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{2}{1} = 2$

To find $u$: $u = \tan^{-1}(2)$

Now, we use double angle formulas to find $\sin(2u)$, $\cos(2u)$, and $\tan(2u)$.

1. $\sin(2u)$

The double angle formula for sine is: $\sin(2u) = 2 \sin(u) \cos(u)$

First, we find $\sin(u)$ and $\cos(u)$:

Using the coordinates of the point $(1, 2)$: $r = \sqrt{x^2 + y^2} = \sqrt{1^2 + 2^2} = \sqrt{5}$

$\sin(u) = \frac{y}{r} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$ $\cos(u) = \frac{x}{r} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$

Now, substituting into the double angle formula for sine: $\sin(2u) = 2 \sin(u) \cos(u) = 2 \left(\frac{2\sqrt{5}}{5}\right) \left(\frac{\sqrt{5}}{5}\right)$ $\sin(2u) = 2 \left(\frac{2\sqrt{5}}{5} \cdot \frac{\sqrt{5}}{5}\right)$ $\sin(2u) = 2 \left(\frac{2 \cdot 5}{25}\right)$ $\sin(2u) = 2 \left(\frac{2}{5}\right)$ $\sin(2u) = \frac{4}{5}$

2. $\cos(2u)$

The double angle formula for cosine is: $\cos(2u) = \cos^2(u) - \sin^2(u)$

Using the values for $\cos(u)$ and $\sin(u)$: $\cos^2(u) = \left(\frac{\sqrt{5}}{5}\right)^2 = \frac{5}{25} = \frac{1}{5}$ $\sin^2(u) = \left(\frac{2\sqrt{5}}{5}\right)^2 = \frac{4 \cdot 5}{25} = \frac{20}{25} = \frac{4}{5}$

Now, substituting into the double angle formula for cosine: $\cos(2u) = \cos^2(u) - \sin^2(u) = \frac{1}{5} - \frac{4}{5}$ $\cos(2u) = \frac{1 - 4}{5}$ $\cos(2u) = -\frac{3}{5}$

3. $\tan(2u)$

The double angle formula for tangent is: $\tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)}$

Using $\tan(u) = 2$: $\tan(2u) = \frac{2 \cdot 2}{1 - 2^2}$ $\tan(2u) = \frac{4}{1 - 4}$ $\tan(2u) = \frac{4}{-3}$ $\tan(2u) = -\frac{4}{3}$

So, the exact values are:

• $\sin(2u) = \frac{4}{5}$
• $\cos(2u) = -\frac{3}{5}$
• $\tan(2u) = -\frac{4}{3}$

Would you like more details or have any questions about these calculations?

Here are 5 more questions you might find interesting:

1. How do you derive the double angle formulas for sine and cosine?
2. Can you explain the unit circle and its importance in trigonometry?
3. What are the sum and difference formulas for sine, cosine, and tangent?
4. How do you convert between degrees and radians?
5. What is the Pythagorean identity in trigonometry?

Tip: When working with trigonometric functions, always make sure to check if you need to work in degrees or radians to avoid errors in calculations.