Find all angles for triangle with the sides a = 5 b=3.9 c= 3.3

To find the angles of a triangle with sides \( a = 5 \), \( b = 3.9 \), and \( c = 3.3 \), we can use the **Law of Cosines**:

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

### Step 1: Calculate Angle \( A \)
Substitute \( a = 5 \), \( b = 3.9 \), and \( c = 3.3 \):

\[
\cos A = \frac{3.9^2 + 3.3^2 - 5^2}{2 \cdot 3.9 \cdot 3.3}
\]

1. \( b^2 = 3.9^2 = 15.21 \)
2. \( c^2 = 3.3^2 = 10.89 \)
3. \( a^2 = 5^2 = 25 \)

\[
\cos A = \frac{15.21 + 10.89 - 25}{2 \cdot 3.9 \cdot 3.3}
\]
\[
\cos A = \frac{1.1}{25.74} \approx 0.0428
\]
\[
A \approx \arccos(0.0428) \approx 87.55^\circ
\]

### Step 2: Calculate Angle \( B \)
\[
\cos B = \frac{5^2 + 3.3^2 - 3.9^2}{2 \cdot 5 \cdot 3.3}
\]

1. \( a^2 = 25 \)
2. \( c^2 = 10.89 \)
3. \( b^2 = 15.21 \)

\[
\cos B = \frac{25 + 10.89 - 15.21}{2 \cdot 5 \cdot 3.3}
\]
\[
\cos B = \frac{20.68}{33} \approx 0.6261
\]
\[
B \approx \arccos(0.6261) \approx 51.14^\circ
\]

### Step 3: Calculate Angle \( C \)
Since the sum of angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - A - B
\]
\[
C = 180^\circ - 87.55^\circ - 51.14^\circ
\]
\[
C \approx 41.31^\circ
\]

### Summary
The angles of the triangle are approximately:
- \( A \approx 87.55^\circ \)
- \( B \approx 51.14^\circ \)
- \( C \approx 41.31^\circ \)

Would you like more details or have any questions?

### Related Questions
1. How does the Law of Cosines apply to triangles in different scenarios?
2. Can this triangle be classified as acute, right, or obtuse?
3. What would happen if one side of the triangle was longer? 
4. How would these angles change if side \( a \) was smaller than both \( b \) and \( c \)?
5. What is the smallest angle that a triangle with given sides can have?

**Tip:** Always remember that the largest angle is opposite the longest side in a triangle!

Find all angles for a triangle with the sides a = 5, b = 3.9, c = 3.3

Explore a step-by-step solution to find the angles of a triangle with sides a = 5, b = 3.9, and c = 3.3 using the Law of Cosines. This solution covers trigonometric applications for triangle properties, suitable for high school students in Grades 9-12.

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