Solve each triangle. See Examples 2 and 3.


A = 112.8°, b = 6.28 m, c = 12.2 m

Verified step by step guidance

1. Identify the given elements of the triangle: angle \(A = 112.8^\circ\), side \(b = 6.28\) m, and side \(c = 12.2\) m. We need to find the remaining angles \(B\) and \(C\), and side \(a\).
2. Use the Law of Cosines to find side \(a\). The Law of Cosines formula is: \[a^2 = b^2 + c^2 - 2bc \cos A\] Substitute the known values of \(b\), \(c\), and \(A\) into this formula.
3. Calculate \(a\) by taking the square root of the result from the Law of Cosines step: \[a = \sqrt{b^2 + c^2 - 2bc \cos A}\]
4. Use the Law of Sines to find one of the unknown angles, for example angle \(B\). The Law of Sines states: \[\frac{\sin A}{a} = \frac{\sin B}{b}\] Rearrange to solve for \(\sin B\): \[\sin B = b \times \frac{\sin A}{a}\]
5. Calculate angle \(B\) by taking the inverse sine (arcsin) of \(\sin B\). Then find angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\): \[C = 180^\circ - A - B\]

Key Concepts

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful for solving triangles when two sides and the included angle are known. The formula is c² = a² + b² - 2ab cos(C), allowing calculation of unknown sides or angles.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing one angle and two sides allows you to find the remaining angles by subtracting the known angles from 180°, which is essential for fully solving the triangle.

Law of Sines

The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant in a triangle: a/sin(A) = b/sin(B) = c/sin(C). This law helps find unknown angles or sides when given an angle-side pair and another side or angle.