Use the figure to find the exat values of.

Sin(2u) = 2sin(u) cos(u)

Sin(2u) =2() () = 2* =
 
Cos2(u) ==

2==

 = -   = -

Tan(2u) =  

 =

Sure, let's use the given formulas and values we found earlier to explicitly calculate \(\sin(2u)\), \(\cos(2u)\), and \(\tan(2u)\).

### 1. \(\sin(2u)\)

Using the double angle formula:
\[ \sin(2u) = 2 \sin(u) \cos(u) \]

We previously found:
\[ \sin(u) = \frac{2\sqrt{5}}{5} \]
\[ \cos(u) = \frac{\sqrt{5}}{5} \]

Now, substituting these values:
\[ \sin(2u) = 2 \left(\frac{2\sqrt{5}}{5}\right) \left(\frac{\sqrt{5}}{5}\right) \]

Calculating the product inside:
\[ \sin(2u) = 2 \left(\frac{2\sqrt{5} \cdot \sqrt{5}}{5 \cdot 5}\right) \]
\[ \sin(2u) = 2 \left(\frac{2 \cdot 5}{25}\right) \]
\[ \sin(2u) = 2 \left(\frac{2}{5}\right) \]
\[ \sin(2u) = \frac{4}{5} \]

So, the exact value of \(\sin(2u)\) is \(\frac{4}{5}\).

### 2. \(\cos(2u)\)

Using the double angle formula:
\[ \cos(2u) = \cos^2(u) - \sin^2(u) \]

We previously found:
\[ \cos^2(u) = \left(\frac{\sqrt{5}}{5}\right)^2 = \frac{5}{25} = \frac{1}{5} \]
\[ \sin^2(u) = \left(\frac{2\sqrt{5}}{5}\right)^2 = \frac{20}{25} = \frac{4}{5} \]

Now, substituting these values:
\[ \cos(2u) = \cos^2(u) - \sin^2(u) \]
\[ \cos(2u) = \frac{1}{5} - \frac{4}{5} \]
\[ \cos(2u) = \frac{1 - 4}{5} \]
\[ \cos(2u) = -\frac{3}{5} \]

So, the exact value of \(\cos(2u)\) is \(-\frac{3}{5}\).

### 3. \(\tan(2u)\)

Using the double angle formula:
\[ \tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)} \]

We previously found:
\[ \tan(u) = 2 \]

Now, substituting this value:
\[ \tan(2u) = \frac{2 \cdot 2}{1 - 2^2} \]
\[ \tan(2u) = \frac{4}{1 - 4} \]
\[ \tan(2u) = \frac{4}{-3} \]
\[ \tan(2u) = -\frac{4}{3} \]

So, the exact value of \(\tan(2u)\) is \(-\frac{4}{3}\).

To summarize, 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 identities and formulas, it is useful to keep a list of common identities handy for quick reference.

Learn how to find the exact values of \( \sin(2u) \), \( \cos(2u) \), and \( \tan(2u) \) using double angle formulas in trigonometry. Detailed step-by-step calculations provided with example values. Suitable for advanced high school students studying trigonometric identities.

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