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Chapter 6: Problem 56

Describe a strategy for solving an SSS triangle.

Short Answer

Expert verified
Use the Law of Cosines with the given side lengths to find one angle. Then, use the Law of Sines to find a second angle. Finally, calculate the third angle using the fact that the sum of the angles within a triangle is 180 degrees.

Step by step solution

01

Use the Law of Cosines to find an angle of the triangle

First use the Law of Cosines to find one of the angles in the triangle. Given a triangle with sides of length a, b, and c, and where \(\gamma\) is the angle opposite the side of length c, you can use the following equation: \[\cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab}\] Solve this equation to find the angle \(\gamma\).
02

Use the Law of Sines to find another angle

Next, use the Law of Sines to find another angle. Given that \(\alpha\) is the angle opposite side a, you can use the following equation: \[\frac{\sin(\alpha)}{a} = \frac{\sin(\gamma)}{c}\] Solve this equation to find angle \(\alpha\).
03

Use the triangle angle sum to find the last angle

Using the fact that the sum of internal angles in a triangle is 180 degrees, you can find the third angle. If \(\beta\) is this angle, then: \[\beta = 180 - \alpha - \gamma\] Solve this equation to find angle \(\beta\).

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Key Concepts

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

Law of Cosines
Understanding the Law of Cosines is essential when dealing with a triangle where all three sides (SSS) are known. Imagine you have a triangle with sides labeled as \(a\), \(b\), and \(c\). You want to find one of the angles opposite to these sides. The Law of Cosines is the perfect tool for this. It relates the length of a triangle鈥檚 sides to the cosine of one of its angles. Here's how it works:
  • The formula is \( \cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab} \), where \(\gamma\) is the angle opposite to side \(c\).
  • This formula is similar to the Pythagorean theorem, but it allows for calculation in any triangle, not just right-angled ones.
  • To find the angle, compute and then solve for \(\gamma\) using an inverse cosine function.
This technique is very useful, especially when dealing with real-world problems, where measuring angles directly can be challenging.
Law of Sines
Once you have one angle from the triangle using the Law of Cosines, you can proceed with the Law of Sines to find another. The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant in any triangle. This is the formula:
  • \( \frac{\sin(\alpha)}{a} = \frac{\sin(\gamma)}{c} \), where \(\alpha\) and \(\gamma\) are angles opposite sides \(a\) and \(c\) respectively.
  • Find \(\alpha\) by solving this proportion. Use the known values of sides and angles to solve for the unknown angle.
This method is truly handy as it allows for the determination of missing angles or sides when at least one side length and its opposite angle are known. Mastering this law opens the door to solving many practical triangle scenarios.
Triangle Angle Sum
The final piece to completing any triangle problem is understanding the triangle angle sum property. The internal angles of a triangle always add up to 180 degrees. This principle is fundamental and applies to all triangles, regardless of their type. Here's how you can use it:
  • Once two angles of the triangle (\(\alpha\) and \(\gamma\)) are known, use \(\beta = 180 - \alpha - \gamma\) to find the third angle \(\beta\).
  • This is straightforward and guarantees that you've found an accurate measure for every angle in the triangle.
This rule is a simple but powerful concept that makes solving triangle problems methodical and logical. It serves as a good check to ensure the angle measures make sense after using the other laws.

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Most popular questions from this chapter

Two ships leave a harbor at the same time. One ship travels on a bearing of \(\mathrm{S} 12^{\circ} \mathrm{W}\) at 14 miles per hour. The other ship travels on a bearing of \(\mathrm{N} 75^{\circ} \mathrm{E}\) at 10 miles per hour. How far apart will the ships be after three hours? Round to the nearest tenth of a mile.

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$A=56^{\circ}, C=24^{\circ}, a=22$$

In converting \(r=5\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

You are on a fishing boat that leaves its pier and heads east. After traveling for 30 miles, there is a report warning of rough seas directly south. The captain turns the boat and follows a bearing of \(\mathrm{S} 45^{\circ} \mathrm{W}\) for 12 miles. a. At this time, how far are you from the boat's pier? Round to the nearest tenth of a mile. b. What bearing could the boat have originally taken to arrive at this spot?

Find the work done in pushing a car along a level road from point \(A\) to point \(B, 80\) feet from \(A,\) while exerting a constant force of 95 pounds. Round to the nearest foot-pound.
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