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

A = 142.13°, b = 5.432 ft, a = 7.297 ft

Verified step by step guidance

Step 1: Use the Law of Sines to find angle B. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the known values: \( \frac{7.297}{\sin 142.13^\circ} = \frac{5.432}{\sin B} \).

Step 2: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{5.432 \cdot \sin 142.13^\circ}{7.297} \).

Step 3: Calculate \( \sin B \) using the value obtained from Step 2.

Step 4: Determine angle B by taking the inverse sine (arcsin) of \( \sin B \).

Step 5: Find angle C using the fact that the sum of angles in a triangle is 180°. Use the equation: \( C = 180^\circ - A - B \).

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Key Concepts

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

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This fundamental principle allows us to find unknown angles when at least one angle is known. In this case, knowing angle A enables us to calculate the other two angles by subtracting A from 180°.

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It is expressed as a/b = sin(A)/sin(B) = c/sin(C). This law is particularly useful in non-right triangles, allowing us to find unknown angles or sides when we have sufficient information, such as two sides and a non-included angle.

Angle of Elevation and Depression

The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line, while the angle of depression is formed when looking down. Understanding these angles is crucial in trigonometry, especially in real-world applications involving heights and distances, as they often require the use of trigonometric ratios to solve for unknown values.

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