Solve each triangle ABC that exists.
A = 38° 40', a = 9.72 m, b = 11.8 m

Verified step by step guidance

1. Convert the given angle A from degrees and minutes to decimal degrees for easier calculation. Recall that 1 minute is \( \frac{1}{60} \) of a degree, so calculate \( A = 38 + \frac{40}{60} \) degrees.
2. Use the Law of Sines to find angle B. The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Rearrange to solve for \( \sin B \): \( \sin B = \frac{b \sin A}{a} \).
3. Calculate \( \sin B \) using the values of \( a \), \( b \), and \( A \) (in decimal degrees), then find angle B by taking the inverse sine (arcsin) of that value.
4. Determine angle C by using the fact that the sum of angles in a triangle is 180 degrees: \( C = 180^\circ - A - B \).
5. Finally, use the Law of Sines again to find side \( c \) with the formula \( \frac{c}{\sin C} = \frac{a}{\sin A} \), rearranged as \( c = \frac{a \sin C}{\sin A} \).

Key Concepts

Law of Sines

The Law of Sines relates the sides and angles of a triangle through the ratio a/sin(A) = b/sin(B) = c/sin(C). It is essential for solving triangles when two sides and an angle are known, allowing calculation of unknown angles or sides.

Triangle Ambiguity (SSA Case)

When two sides and a non-included angle (SSA) are given, there may be zero, one, or two possible triangles. Understanding this ambiguity helps determine if the triangle exists and how many solutions are possible.

Angle Conversion and Notation

Angles given in degrees and minutes must be converted to decimal degrees or radians for calculations. Accurate conversion ensures precise use of trigonometric functions and correct problem-solving.