Find the unknown angles in triangle ABC for each triangle that exists.


A = 29.7°, b = 41.5 ft, a = 27.2 ft

Verified step by step guidance

1. Identify the given elements: angle \(A = 29.7^\circ\), side \(a = 27.2\) ft (opposite angle \(A\)), and side \(b = 41.5\) ft (opposite angle \(B\)). We need to find the unknown angles \(B\) and \(C\) in triangle \(ABC\).
2. Use the Law of Sines to find angle \(B\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearranged to solve for \(\sin B\), it becomes \(\sin B = \frac{b \cdot \sin A}{a}\).
3. Calculate \(\sin B\) using the known values, then find angle \(B\) by taking the inverse sine (arcsin) of that value. Remember that the sine function can have two possible angles between \(0^\circ\) and \(180^\circ\), so consider both possible values for \(B\) if applicable.
4. For each possible value of angle \(B\), find angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\). So, \(C = 180^\circ - A - B\).
5. Check the validity of each triangle by ensuring all angles are positive and the sides satisfy the triangle inequality. This will confirm how many triangles exist with the given data.

Key Concepts

Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is essential for finding unknown angles or sides when given two sides and an angle not included between them.

Ambiguous Case of the Law of Sines (SSA Condition)

When two sides and a non-included angle (SSA) are known, there can be zero, one, or two possible triangles. This ambiguity arises because the given data may correspond to different configurations, so checking for the number of valid solutions is crucial.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. After finding one unknown angle using the Law of Sines, this property helps determine the remaining angle by subtracting the known angles from 180°.