Solve each triangle ABC that exists.
A = 96.80°, b = 3.589 ft, a = 5.818 ft

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Identify the given information: angle \( A = 96.80^\circ \), side \( b = 3.589 \) ft, and side \( a = 5.818 \) ft.

Use the Law of Sines to find angle \( B \): \( \frac{\sin A}{a} = \frac{\sin B}{b} \).

Rearrange the equation to solve for \( \sin B \): \( \sin B = \frac{b \cdot \sin A}{a} \).

Calculate \( \sin B \) using the values of \( A, a, \) and \( b \).

Use the inverse sine function to find angle \( B \), and then find angle \( C \) using \( C = 180^\circ - A - B \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle ABC, the ratio of a side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This law is particularly useful for solving triangles when given two angles and one side or two sides and a non-included angle.

Triangle Properties

Understanding the properties of triangles is essential for solving them. A triangle's angles always sum to 180 degrees, which allows for the calculation of unknown angles when some are known. Additionally, the relationship between the sides and angles, such as the longest side being opposite the largest angle, is crucial for applying trigonometric laws effectively.

Angle Measurement

In trigonometry, angles can be measured in degrees or radians, and it's important to be consistent with the unit used throughout calculations. The given angle A = 96.80° indicates that the triangle is an obtuse triangle, which affects the possible configurations of the other angles and sides. Understanding how to convert between degrees and radians can also be beneficial when applying trigonometric functions.

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