Solve each triangle ABC.


B = 38° 40', a = 19.7 cm, C = 91° 40'

Verified step by step guidance

1. Step 1: Start by finding angle A using the fact that the sum of angles in a triangle is 180°. Use the formula: A = 180° - B - C.
2. Step 2: Convert the given angles B and C from degrees and minutes to decimal degrees if necessary for easier calculations.
3. Step 3: Calculate angle A using the values of B and C. Substitute B = 38° 40' and C = 91° 40' into the formula from Step 1.
4. Step 4: Use the Law of Sines to find side b. The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Rearrange to solve for b: \( b = a \cdot \frac{\sin B}{\sin A} \).
5. Step 5: Similarly, use the Law of Sines to find side c. Use the formula: \( c = a \cdot \frac{\sin C}{\sin A} \).

Key Concepts

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. In this problem, knowing two angles allows us to calculate the third angle, which is essential for solving the triangle. This theorem is fundamental in trigonometry as it helps establish relationships between the angles and sides of triangles.

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful in non-right triangles, allowing us to find unknown side lengths or angles when given sufficient information.

Angle Conversion

In trigonometry, angles can be expressed in degrees or radians, and it is often necessary to convert between these two units. In this problem, the angles are given in degrees and minutes, which must be converted to decimal degrees for calculations. Understanding how to perform these conversions is crucial for accurately applying trigonometric principles.