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Chapter 5: Problem 71

Solve the triangle with \(\alpha=108.1^{\circ}, \beta=18.6^{\circ},\) and \(c=28.6\).

Short Answer

Expert verified
The third angle is 53.3掳; side lengths are a \approx 35.68 and b \approx 10.54.

Step by step solution

01

- Find the Third Angle

To solve for the missing angle \(\theta\), use the fact that the sum of all angles in a triangle is 180 degrees. Therefore, \(\theta = 180^\text{掳} - \alpha - \beta\). Compute \(\theta = 180^\text{掳} - 108.1^\text{掳} - 18.6^\text{掳} = 53.3^\text{掳}\).
02

- Apply the Law of Sines

Use the Law of Sines to solve for the sides \(a\) and \(b\). The Law of Sines states: i\(\frac{a}{\text{sin}(\theta_a)} = \frac{b}{\text{sin}(\theta_b)} = \frac{c}{\text{sin}(\theta_c)}\)i Since we know \(c=28.6\), \(\alpha=108.1^\text{掳}\), \(\beta=18.6^\text{掳}\), and \(\theta=53.3^\text{掳}\), we can set up the equation: i\(\frac{a}{\text{sin}(108.1^\text{掳})} = \frac{28.6}{\text{sin}(53.3^\text{掳})}\)i\(\frac{b}{\text{sin}(18.6^\text{掳})} = \frac{28.6}{\text{sin}(53.3^\text{掳})}\)i
03

- Solve for Side a

Isolate \(a\) and solve: i\(a = \frac{28.6 \times \text{sin}(108.1^\text{掳})}{\text{sin}(53.3^\text{掳})}\)i Perform the calculations to find \(a\). i\(a \approx 35.68\)i
04

- Solve for Side b

Isolate \(b\) and solve: i\(b = \frac{28.6 \times \text{sin}(18.6^\text{掳})}{\text{sin}(53.3^\text{掳})}\)i Perform the calculations to find \(b\). i\(b \approx 10.54\)i

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometry
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. In a right triangle, the primary trigonometric functions are sine, cosine, and tangent, but these concepts extend into general triangles through laws like the Law of Sines and Cosines. In any triangle, you can use trigonometric principles to find unknown angles and sides when you have initial information, such as one side and two angles, or two sides and one angle. In this exercise, trigonometry helps us systematically find the missing side lengths and angles of the given triangle.
Law of Sines
The Law of Sines is a fundamental rule in trigonometry that applies to any triangle. It states that the ratio of the length of a side of a triangle to the sine of the opposite angle is the same for all three sides and angles of the triangle. The law is expressed as: \( \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)} \), where \(a, b,\) and \(c\) are the lengths of the sides, and \(A, B,\) and \(C\) are the angles opposite those sides. In our exercise, after determining the missing angle using the angle sum property, we utilized the Law of Sines to find the unknown side lengths \(a\) and \(b\). This involved setting up and solving equations based on known values of the given side length \(c\) and its corresponding angle.
Angle Sum Property
The angle sum property is a simple yet vital concept in geometry. It states that the sum of the interior angles of any triangle is always 180 degrees. This concept is essential for solving triangles because it allows us to find a missing angle if the other two angles are known. In our exercise, we used the angle sum property to find the third angle \(\theta\), since we were given \(\alpha = 108.1^\text{掳}\) and \ (\beta = 18.6^\text{掳}. By subtracting the sum of these angles from 180 degrees, we determined that \(\theta = 53.3^\text{掳}\). This newfound angle was crucial for applying the Law of Sines to solve for the unknown sides.

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