 Calculate all unknown sides and angles for the triangle ABC, given A = 57°, B = 72° and a = 48.2.


To solve for all the unknown sides and angles of triangle ABC, we are given:

- \( A = 57^\circ \)
- \( B = 72^\circ \)
- \( a = 48.2 \)

### Step 1: Find Angle C
The sum of angles in any triangle is \( 180^\circ \). Therefore, we can find angle \( C \) as follows:

\[
C = 180^\circ - A - B = 180^\circ - 57^\circ - 72^\circ = 51^\circ
\]

### Step 2: Use the Law of Sines to Find the Unknown Sides
The Law of Sines states:

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

We can use this to find sides \( b \) and \( c \).

#### Find side \( b \):

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

\[
b = \frac{a \cdot \sin B}{\sin A} = \frac{48.2 \cdot \sin(72^\circ)}{\sin(57^\circ)}
\]

#### Find side \( c \):

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

\[
c = \frac{a \cdot \sin C}{\sin A} = \frac{48.2 \cdot \sin(51^\circ)}{\sin(57^\circ)}
\]

Let's calculate these values.The calculations yield the following results:

- **Angle \( C \)**: \( 51^\circ \)
- **Side \( b \)**: \( 54.66 \) (rounded to two decimal places)
- **Side \( c \)**: \( 44.66 \) (rounded to two decimal places)

Would you like more details on the process or have any further questions?

### Related Questions:
1. How would the Law of Cosines be applied to this problem?
2. What are the properties of the triangle if angle \( C \) were larger than \( 90^\circ \)?
3. Can we solve this problem using only the Law of Cosines? If so, how?
4. How does changing side \( a \) affect the other sides and angles?
5. How would you find the area of this triangle using the given information?

### Tip:
Always double-check your angle calculations in trigonometry, as small errors can lead to significant discrepancies in side lengths.

Learn how to calculate all unknown sides and angles of triangle ABC given angle A = 57°, angle B = 72°, and side a = 48.2. Detailed step-by-step solution using trigonometric principles and the Law of Sines. Suitable for high school students studying trigonometry.

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