Solve each triangle ABC.


B = 42.88°, C = 102.40°, b = 3974 ft

Verified step by step guidance

1. First, find the measure of angle A using the fact that the sum of the angles in any triangle is 180°. Use the formula: \(A = 180^\circ - B - C\).
2. Next, apply the Law of Sines to find side \(a\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearrange to solve for \(a\): \(a = b \times \frac{\sin A}{\sin B}\).
3. Then, use the Law of Sines again to find side \(c\) with the formula: \(\frac{c}{\sin C} = \frac{b}{\sin B}\). Rearrange to solve for \(c\): \(c = b \times \frac{\sin C}{\sin B}\).
4. At this point, you have all three angles (\(A\), \(B\), \(C\)) and all three sides (\(a\), \(b\), \(c\)) of triangle ABC determined.
5. Finally, review your answers to ensure all sides and angles are consistent and make sense within the triangle, remembering that side lengths must be positive and angles must sum to 180°.

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 an angle-side pair and another angle or side.

Solving Triangles

Solving a triangle means finding all unknown sides and angles. Using known angles and sides, along with the Law of Sines and angle sum property, allows you to determine the missing measurements accurately.