Solve each triangle ABC.


B = 38° 40', a = 19.7 cm, C = 91° 40'

Verified step by step guidance

1. First, convert the given angles from degrees and minutes to decimal degrees for easier calculation. For example, 38° 40' becomes \(38 + \frac{40}{60}\) degrees, and similarly for 91° 40'.
2. Next, find the measure of the third angle \(A\) in triangle \(ABC\) using the fact that the sum of angles in a triangle is 180°. So, calculate \(A = 180° - B - C\).
3. Use the Law of Sines to find the missing sides. The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). You already know side \(a\) and angles \(B\) and \(C\), and now angle \(A\).
4. Set up the ratio to find side \(b\): \(b = a \times \frac{\sin B}{\sin A}\). Calculate \(b\) using the known values.
5. Similarly, find side \(c\) using \(c = a \times \frac{\sin C}{\sin A}\). This will give you all sides and angles of triangle \(ABC\).

Key Concepts

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third by subtracting their sum from 180°, which is essential for solving the triangle completely.

Law of Sines

The Law of Sines relates the sides and angles of a triangle: (a/sin A) = (b/sin B) = (c/sin C). It is used to find unknown sides or angles when given a combination of sides and angles, especially in non-right triangles.

Angle Conversion and Notation

Angles given in degrees and minutes must be converted to decimal degrees or radians for calculation. Understanding how to interpret and convert these units ensures accurate use of trigonometric formulas.