In ΔABC, a = 7.4 cm, b = 4.6 cm and c=8.6 cm. Find the measure of ∠A to the nearest 10th of a degree.

To find the measure of \( \angle A \) in \( \triangle ABC \) with sides \( a = 7.4 \, \text{cm} \), \( b = 4.6 \, \text{cm} \), and \( c = 8.6 \, \text{cm} \), we can use the **Cosine Rule**:

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]

### Step-by-Step Solution

1. **Substitute the values**:
   \[
   \cos A = \frac{(4.6)^2 + (8.6)^2 - (7.4)^2}{2 \cdot 4.6 \cdot 8.6}
   \]

2. **Calculate each term**:
   - \( b^2 = 4.6^2 = 21.16 \)
   - \( c^2 = 8.6^2 = 73.96 \)
   - \( a^2 = 7.4^2 = 54.76 \)

3. **Substitute and simplify**:
   \[
   \cos A = \frac{21.16 + 73.96 - 54.76}{2 \cdot 4.6 \cdot 8.6}
   \]
   \[
   \cos A = \frac{40.36}{79.12}
   \]
   \[
   \cos A \approx 0.51
   \]

4. **Find \( \angle A \) using the inverse cosine**:
   \[
   A = \cos^{-1}(0.51) \approx 59.4^\circ
   \]

### Answer
The measure of \( \angle A \) is approximately **59.4°**.

Would you like further details or have any questions?

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**Related Questions:**

1. How would you find \( \angle B \) using the same side lengths?
2. What if you needed to solve for \( \angle C \) in \( \triangle ABC \)?
3. How does the Cosine Rule change if solving for a side length instead of an angle?
4. How could the Sine Rule be used to find \( \angle A \) after finding one angle?
5. What steps are needed to verify if this is a valid triangle?

**Tip:** The Cosine Rule is especially useful when you know all three side lengths of a triangle and want to find an angle.

In ΔABC, a = 7.4 cm, b = 4.6 cm and c = 8.6 cm. Find the measure of ∠A to the nearest 10th of a degree.

Explore a step-by-step solution for finding the measure of angle A in triangle ABC using the Cosine Rule. Given side lengths a = 7.4 cm, b = 4.6 cm, and c = 8.6 cm, this trigonometry problem demonstrates key geometric concepts and calculations. Ideal for students in Grades 10-12.

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