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To solve for the angle \( w \) in the triangle, we can apply the **law of cosines**. The law of cosines states that:

\[
\cos(w) = \frac{a^2 + b^2 - c^2}{2ab}
\]

where:
- \( a = 22 \)
- \( b = 34 \)
- \( c = 15 \) (opposite the angle \( w \))

Substituting the values into the equation, we can solve for \( w \). Let me calculate that for you.The value of angle \( w \) is approximately \( 18.94^\circ \) when rounded to two decimal places.

Would you like more details on the calculation, or do you have any further questions?

Here are 5 related questions to expand your understanding:
1. How does the Law of Cosines work in non-right triangles?
2. Can you use the Law of Sines to find other angles in this triangle?
3. What would happen if the triangle were a right triangle—how would the methods change?
4. How can the Law of Cosines be applied to real-world problems?
5. What are the steps for verifying this result using a different method?

**Tip**: When working with angles in trigonometry, always ensure your calculator is set to the correct mode (degrees or radians) based on the problem context.

Find the value of angle w in degrees. The triangle has sides of lengths 22, 34, and 15 with angle w opposite the side of length 15.

Learn how to use the Law of Cosines to find the value of angle w in a triangle with sides of lengths 22, 34, and 15. This problem provides a step-by-step solution, including key trigonometric concepts and formulas. Suitable for students in Grades 10-12.

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