Find all the side lengths, correct to 2 decimal places, for the triangle ABC, given a=10.5, B=60∘ and C=72∘.

To find all the side lengths in triangle \( ABC \), we can use the Law of Sines. The given data includes:
- Side \( a = 10.5 \)
- Angle \( B = 60^\circ \)
- Angle \( C = 72^\circ \)

First, we need to find angle \( A \) using the fact that the sum of angles in a triangle is \( 180^\circ \):

\[
A = 180^\circ - B - C
\]

After calculating angle \( A \), we can apply the Law of Sines:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We'll solve for sides \( b \) and \( c \). Let's proceed with the calculations.The side lengths for triangle \( ABC \) are:
- \( b \approx 12.24 \)
- \( c \approx 13.44 \)

These values are rounded to two decimal places. If you need further explanations or have any questions, please let me know!

Here are some related questions you might find interesting:
1. How can we verify the calculations using the Law of Cosines?
2. What is the significance of the Law of Sines in solving triangles?
3. How can we determine the area of triangle \( ABC \) with the given sides and angles?
4. What would change if triangle \( ABC \) were a right triangle?
5. How can we find the circumradius of triangle \( ABC \)?
6. How does knowing all sides and angles help in real-world applications?
7. What are some other trigonometric rules that can solve triangles?
8. Can the Law of Sines be applied to obtuse triangles?

**Tip:** Always ensure your calculator is set to the correct mode (degrees or radians) when working with trigonometric functions.

Learn how to find all the side lengths of triangle ABC using the Law of Sines. Given a=10.5, B=60°, and C=72°, calculate sides b and c approximately to two decimal places. Explore the significance of the Law of Sines in triangle geometry.

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