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Chapter 9: Problem 14

Solve the triangles with the given parts. $$a=0.130, A=55.2^{\circ}, B=117.5^{\circ}$$

Short Answer

Expert verified
The triangle has sides \( a = 0.130 \), \( b \approx 0.150 \), \( c \approx 0.018 \) and angles \( A = 55.2^{\circ} \), \( B = 117.5^{\circ} \), \( C = 7.3^{\circ} \).

Step by step solution

01

Determine Angle C

In any triangle, the sum of all interior angles is always 180 degrees. Given that angles A and B are: \( A = 55.2^{\circ} \) and \( B = 117.5^{\circ} \), we can find angle C using the formula: \[ C = 180^{\circ} - A - B \]Substitute the known values:\[ C = 180^{\circ} - 55.2^{\circ} - 117.5^{\circ} = 7.3^{\circ} \]
02

Use the Law of Sines to Find Side b

The Law of Sines states:\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \]Where we need to find side \( b \). Given \( a = 0.130 \), \( A = 55.2^{\circ} \), and \( B = 117.5^{\circ} \), substitute the values:\[ \frac{0.130}{\sin(55.2^{\circ})} = \frac{b}{\sin(117.5^{\circ})} \]Solving for \( b \):\[ b = \frac{0.130 \times \sin(117.5^{\circ})}{\sin(55.2^{\circ})} \]Calculate using a calculator:\[ b \approx 0.150 \]
03

Use the Law of Sines to Find Side c

Using the Law of Sines again with the relationship:\[ \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \]Given \( a = 0.130 \), \( A = 55.2^{\circ} \), and \( C = 7.3^{\circ} \), substitute the values:\[ \frac{0.130}{\sin(55.2^{\circ})} = \frac{c}{\sin(7.3^{\circ})} \]Solving for \( c \):\[ c = \frac{0.130 \times \sin(7.3^{\circ})}{\sin(55.2^{\circ})} \]Calculate using a calculator:\[ c \approx 0.018 \]

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Key Concepts

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

Angles of a Triangle
Triangles are polygons that have three sides and three angles. The angles of a triangle are always essential in determining its shape and size. Whether you're dealing with an equilateral, isosceles, or scalene triangle, knowing the properties of angles can help solve various problems.
Each angle in a triangle is formed by the intersection of two sides. In simple terms, angles help us understand the corners of the triangle:
  • Acute Angles: Each angle is less than 90 degrees.
  • Right Angle: One angle is exactly 90 degrees.
  • Obtuse Angle: One angle is greater than 90 degrees.
By understanding these types, you're better equipped to approach each triangle problem methodically.
Law of Sines
The Law of Sines is a useful tool in trigonometry to solve triangle problems when dealing with non-right angles. It's particularly helpful in triangles where traditional methods, like the Pythagoras theorem, don't apply. The formula is:
  • \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)
This formula ties together the sides and angles of any triangle. Here's how you can use it:
  • Identify which sides and angles you're given.
  • Substitute the known values into the formula.
  • Solve for the unknown sides or angles.
Learning to use this tool effectively can greatly simplify the process of solving triangles.
Interior Angles Sum
A crucial property of triangles is that the sum of their interior angles is always 180 degrees. This principle applies to every type of triangle, regardless of their side lengths or shapes.
To find a missing angle, simply subtract the sum of the known angles from 180 degrees:
  • Formula: \( C = 180^\circ - A - B \)
Understanding the sum of interior angles helps to quickly identify or verify the angles of a triangle. It provides a foundational check ensuring that the calculation of any triangle's angles is correct. This principle is not only foundational but also very practical in everyday problem-solving scenarios.

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