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Magazine Chapter 14: Problem 18
In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, a=22, b=18,\) and \(\mathrm{m} \angle C=130\)
Short Answer
Expert verified
c 鈮� 34, A 鈮� 23掳, B 鈮� 27掳.
Step by step solution
01
Determine the Type of Triangle Problem
This problem involves solving a triangle given two sides and an included angle. Hence, this is an ASA (Angle-Side-Angle) problem.
02
Apply the Law of Cosines to Find Side c
Use the Law of Cosines to find the length of side \( c \):\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Substitute \( a = 22 \), \( b = 18 \), and \( C = 130^\circ \):\[c^2 = 22^2 + 18^2 - 2 \cdot 22 \cdot 18 \cdot \cos(130^\circ) \]Calculate \( c^2 \) and then take the square root to find \( c \):\[c \approx 34 \] (rounded to the nearest integer).
03
Apply the Law of Sines to Find Angle A
Use the Law of Sines:\[\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\]Substitute \( a = 22 \), \( c \approx 34 \), and \( C = 130^\circ \):\[\frac{22}{\sin(A)} = \frac{34}{\sin(130^\circ)}\]Solve for \( \sin(A) \), and then find \( A \) using the inverse sine function:\[A \approx 23^\circ \] (rounded to the nearest degree).
04
Find Angle B Using the Sum of Angles in a Triangle
Use the fact that the sum of angles in a triangle is 180 degrees:\[A + B + C = 180^\circ \]Substitute the known values: \( A \approx 23^\circ \) and \( C = 130^\circ \):\[23^\circ + B + 130^\circ = 180^\circ\]Solve for \( B \):\[B \approx 27^\circ \] (rounded to the nearest degree).
05
Compile the Triangle Solution
The triangle is solved with the following measurements:- Side \( c \approx 34 \)- Angle \( A \approx 23^\circ \)- Angle \( B \approx 27^\circ \)- Angle \( C = 130^\circ \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Law of Cosines
The Law of Cosines is a fundamental tool in trigonometry that helps to solve triangles, especially when you have two sides and an included angle. It's like a bridge between the sides and angles of a triangle, allowing us to connect and find missing information.
The formula is: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]
We use it when we know two sides of a triangle and the angle between them (like in ASA problems). It's especially helpful for non-right triangles.
By substituting the known values, and solving for the unknown side, you can find its length. Then, by taking the square root, you get the exact measurement needed for further calculations. It鈥檚 a beautiful way to tackle problems in oblique triangles (triangles without a right angle).
The formula is: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]
We use it when we know two sides of a triangle and the angle between them (like in ASA problems). It's especially helpful for non-right triangles.
By substituting the known values, and solving for the unknown side, you can find its length. Then, by taking the square root, you get the exact measurement needed for further calculations. It鈥檚 a beautiful way to tackle problems in oblique triangles (triangles without a right angle).
ASA Triangle
An ASA triangle is a type of triangle where you know two angles and the side between them. It stands for "Angle-Side-Angle." Solving such triangles is a common task, especially in geometry and trigonometry.
In these triangles, you often use the Law of Sines or the Law of Cosines to find missing sides or angles. Given the two angles and the included side, ASA problems are straightforward and systematic.
Recognizing that you're dealing with an ASA triangle allows you to approach the problem with specific strategies, saving time and making the solving process more efficient. The steps involve calculating the third angle using the sum of angles in a triangle and then applying trigonometric laws to find remaining sides.
In these triangles, you often use the Law of Sines or the Law of Cosines to find missing sides or angles. Given the two angles and the included side, ASA problems are straightforward and systematic.
Recognizing that you're dealing with an ASA triangle allows you to approach the problem with specific strategies, saving time and making the solving process more efficient. The steps involve calculating the third angle using the sum of angles in a triangle and then applying trigonometric laws to find remaining sides.
Triangle Solving
Triangle solving is a process of finding unknown sides and angles of a triangle. It's an essential skill in geometry and trigonometry. Each triangle has certain properties and using the right formulas is key to solving them.
The steps include:
Practicing different types of triangle problems improves your problem-solving skills and helps in understanding broader mathematical concepts. Always verify your answers by cross-referencing with known triangle properties.
The steps include:
- Identify what type of triangle you are dealing with (ASA, SSS, etc.).
- Use appropriate trigonometric laws and formulas, such as the Law of Sines or Cosines.
- Find the missing sides and angles systematically.
- Check your solution to ensure all triangle conditions (like sum of angles) are satisfied.
Practicing different types of triangle problems improves your problem-solving skills and helps in understanding broader mathematical concepts. Always verify your answers by cross-referencing with known triangle properties.
Sum of Angles in a Triangle
One of the most fundamental properties of triangles is that the sum of their interior angles is always 180 degrees. This fact is useful because, with just two known angles, you can easily find the third.
The equation for any triangle is: \[A + B + C = 180^\circ\]
This principle applies to all triangles, no matter their type (right, obtuse, or acute).
By using this in conjunction with other laws like Cosines or Sines, you can solve for unknown parts of the triangle. It's a simple yet powerful concept that acts as a check for the correctness of your triangle solutions.
The equation for any triangle is: \[A + B + C = 180^\circ\]
This principle applies to all triangles, no matter their type (right, obtuse, or acute).
By using this in conjunction with other laws like Cosines or Sines, you can solve for unknown parts of the triangle. It's a simple yet powerful concept that acts as a check for the correctness of your triangle solutions.
