Solve each triangle ABC.


C = 79° 18', c = 39.81 mm, A = 32° 57'

Verified step by step guidance

1. Convert the given angles from degrees and minutes to decimal degrees for easier calculations. For example, \( C = 79° 18' \) becomes \( 79 + \frac{18}{60} \) degrees.
2. Use the angle sum property of triangles to find angle \( B \). Since the sum of angles in a triangle is \( 180° \), calculate \( B = 180° - A - C \).
3. Apply the Law of Sines to find side \( a \). The Law of Sines states \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Rearrange to solve for \( a \): \( a = c \cdot \frac{\sin A}{\sin C} \).
4. Use the Law of Sines again to find side \( b \). Similarly, \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Rearrange to solve for \( b \): \( b = c \cdot \frac{\sin B}{\sin C} \).
5. Verify the solution by checking that the calculated sides and angles satisfy the triangle inequality theorem and the sum of angles is \( 180° \).

Key Concepts

Law of Sines

The Law of Sines states that the ratios of the lengths of the sides of a triangle to the sines of their opposite angles are equal. This principle is crucial for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It allows for the calculation of unknown angles and sides, facilitating the solution of triangle ABC in the given problem.

Angle Measurement

Understanding angle measurement is essential in trigonometry, particularly when dealing with degrees and minutes. In the problem, angle C is given as 79° 18', which combines degrees and minutes. Converting this to a decimal format (79.3°) may be necessary for calculations, ensuring accuracy when applying trigonometric functions or the Law of Sines.

Triangle Properties

Triangles have specific properties that govern their angles and sides, notably that the sum of the interior angles in any triangle is always 180°. In this problem, knowing two angles allows for the calculation of the third angle, which is vital for applying the Law of Sines effectively. This foundational property is key to solving triangle ABC.