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To solve this problem, let's break down each part based on the right triangle where \( b = 2 \), \( c = 7 \), and we need to find side \( a \) along with various trigonometric functions.

1. **Finding \( a \) (part a)**:
   Since it's a right triangle, we can use the Pythagorean theorem:
   \[
   a^2 + b^2 = c^2
   \]
   Plugging in the values for \( b \) and \( c \):
   \[
   a^2 + 2^2 = 7^2
   \]
   \[
   a^2 + 4 = 49
   \]
   \[
   a^2 = 45
   \]
   \[
   a = \sqrt{45} = 3\sqrt{5} \approx 6.71
   \]

2. **Calculating trigonometric functions for \( u \)**:
   \( u \) is the angle opposite side \( b \) and adjacent to side \( a \).

   - **\( \cos u \) (part b)**:
     \[
     \cos u = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c} = \frac{3\sqrt{5}}{7} \approx 0.857
     \]

   - **\( \sin u \) (part c)**:
     \[
     \sin u = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} = \frac{2}{7} \approx 0.286
     \]

   - **\( \tan u \) (part d)**:
     \[
     \tan u = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a} = \frac{2}{3\sqrt{5}} = \frac{2\sqrt{5}}{15} \approx 0.298
     \]

3. **Calculating trigonometric functions for \( v \)**:
   \( v \) is the angle opposite side \( a \) and adjacent to side \( b \).

   - **\( \cos v \) (part e)**:
     \[
     \cos v = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} = \frac{2}{7} \approx 0.286
     \]

   - **\( \sin v \) (part f)**:
     \[
     \sin v = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{3\sqrt{5}}{7} \approx 0.857
     \]

   - **\( \tan v \) (part g)**:
     \[
     \tan v = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{3\sqrt{5}}{2} \approx 3.354
     \]

### Summary of Solutions:
- **(a)** \( a = 3\sqrt{5} \approx 6.71 \)
- **(b)** \( \cos u \approx 0.857 \)
- **(c)** \( \sin u \approx 0.286 \)
- **(d)** \( \tan u \approx 0.298 \)
- **(e)** \( \cos v \approx 0.286 \)
- **(f)** \( \sin v \approx 0.857 \)
- **(g)** \( \tan v \approx 3.354 \)

Would you like more details on any of these calculations or have additional questions?

### Related Questions:
1. What would be the values if \( b \) and \( c \) had different values?
2. How would you find the values of \( \cot u \), \( \sec u \), and \( \csc u \)?
3. Can you determine the angle measures \( u \) and \( v \) in degrees?
4. How would the values change if this was a non-right triangle?
5. Can you generalize this method for any right triangle?

### Tip:
For quick estimates, remember that the sine and cosine of complementary angles are equal, i.e., \( \sin u = \cos v \) and \( \cos u = \sin v \).

Suppose b = 2 and c = 7. Evaluate (a) a, (b) cos u, (c) sin u, (d) tan u, (e) cos v, (f) sin v, (g) tan v.

Evaluate trigonometric ratios for a right triangle with sides b = 2 and c = 7. This includes solving for the missing side, a, using the Pythagorean theorem and calculating cos, sin, and tan for angles u and v. Suitable for Grade 9-10 students studying trigonometry.

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