In ΔJKL, k = 710 inches, l = 740 inches and ∠J=17°. Find the length of j, to the nearest inch.

To find the length of side \( j \) in \( \triangle JKL \), given:

- \( k = 710 \) inches
- \( l = 740 \) inches
- \( \angle J = 17^\circ \)

we can use the **Law of Cosines**, which states:
\[
j^2 = k^2 + l^2 - 2kl \cdot \cos(\angle J)
\]

### Step-by-Step Solution

1. Substitute the values into the formula:
   \[
   j^2 = 710^2 + 740^2 - 2 \cdot 710 \cdot 740 \cdot \cos(17^\circ)
   \]

2. Calculate each term:

   - \( 710^2 = 504100 \)
   - \( 740^2 = 547600 \)
   - \( 2 \cdot 710 \cdot 740 = 1050800 \)
   - \( \cos(17^\circ) \approx 0.9563 \)

3. Substitute these results into the equation:
   \[
   j^2 = 504100 + 547600 - 1050800 \cdot 0.9563
   \]

4. Compute \( 1050800 \cdot 0.9563 \approx 1004784 \), so:
   \[
   j^2 = 504100 + 547600 - 1004784
   \]

5. Simplify:
   \[
   j^2 = 515916
   \]

6. Take the square root to find \( j \):
   \[
   j \approx \sqrt{515916} \approx 718
   \]

### Final Answer
The length of \( j \) is approximately **718 inches**.

Would you like a more detailed breakdown, or do you have any other questions?

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1. What is the Law of Cosines and when is it used?
2. How do you find the cosine of an angle without a calculator?
3. What happens if the angle is larger, like \( 90^\circ \)?
4. How would we solve this if we only had two sides and an angle not between them?
5. How do rounding conventions affect accuracy in trigonometric problems?

**Tip:** When using the Law of Cosines, always check that the angle is opposite the side you're solving for.

Explore how to find the length of side j in ΔJKL using the Law of Cosines. Given sides k = 710 inches, l = 740 inches, and ∠J = 17°, this solution provides a step-by-step approach to solving for j. Ideal for students in Grades 10-12 studying trigonometry.

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