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Use the figure to find the exat values of sin2u,cos2u,and tan2u.

To find the exact values of \(\sin(2u)\), \(\cos(2u)\), and \(\tan(2u)\) using the given figure, we need to follow these steps:

1. Determine the coordinates of the point \((1, 2)\) and use them to find \(\sin(u)\) and \(\cos(u)\).
2. Use the double-angle identities to find \(\sin(2u)\), \(\cos(2u)\), and \(\tan(2u)\).

### Step 1: Find \(\sin(u)\) and \(\cos(u)\)

Given the point \((1, 2)\), we can use it to determine \(\sin(u)\) and \(\cos(u)\). The point \((1, 2)\) represents a point on the terminal side of the angle \(u\).

- The length of the hypotenuse \(r\) can be found using the Pythagorean theorem:
  \[
  r = \sqrt{x^2 + y^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
  \]

- Therefore:
  \[
  \cos(u) = \frac{x}{r} = \frac{1}{\sqrt{5}}
  \]
  \[
  \sin(u) = \frac{y}{r} = \frac{2}{\sqrt{5}}
  \]

### Step 2: Use Double-Angle Identities

1. **\(\sin(2u)\)**:
   \[
   \sin(2u) = 2 \sin(u) \cos(u)
   \]
   Substituting the values:
   \[
   \sin(2u) = 2 \left(\frac{2}{\sqrt{5}}\right) \left(\frac{1}{\sqrt{5}}\right) = 2 \cdot \frac{2}{5} = \frac{4}{5}
   \]

2. **\(\cos(2u)\)**:
   \[
   \cos(2u) = \cos^2(u) - \sin^2(u)
   \]
   Substituting the values:
   \[
   \cos^2(u) = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5}
   \]
   \[
   \sin^2(u) = \left(\frac{2}{\sqrt{5}}\right)^2 = \frac{4}{5}
   \]
   \[
   \cos(2u) = \frac{1}{5} - \frac{4}{5} = -\frac{3}{5}
   \]

3. **\(\tan(2u)\)**:
   \[
   \tan(2u) = \frac{\sin(2u)}{\cos(2u)}
   \]
   Substituting the values:
   \[
   \tan(2u) = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}
   \]

### Summary

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

Do you want details or have any questions? Here are 5 questions you might ask:

1. How do we derive the double-angle identities?
2. Can you explain the Pythagorean theorem in more detail?
3. What are other methods to find trigonometric values for given points?
4. How do these identities apply to different quadrants?
5. Can we verify these results using a different approach?

**Tip:** Always ensure to simplify the expressions for \(\sin(u)\) and \(\cos(u)\) before substituting them into the double-angle formulas to avoid calculation errors.

Learn how to find the exact values of sin(2u), cos(2u), and tan(2u) using trigonometric identities and the Pythagorean theorem. Detailed step-by-step solution provided with calculations for a given point (1, 2). Suitable for high school students studying trigonometry.

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