Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Assume triangle \(A B C\) has standard labeling and complete the following. (a) Determine whether SAA, ASA, SSA. SAS, or SSS is given. (b) Decide whether the law of sines or the law of cosines should be used to begin solving the triangle. \(b, c,\) and \(A\)

Step-by-Step Solution

Answer

(a) SAS is given; (b) Use the law of cosines.

Step 1: Identify the Given Information

In triangle \(ABC\), we are given sides \(b\), \(c\), and angle \(A\). We need to determine which configuration this information represents.

Step 2: Determine the Triangle Configuration

With sides \(b\) and \(c\), and angle \(A\) given, the corresponding triangle configuration is Side-Angle-Side (SAS) because we have two sides and the included angle.

Step 3: Decide on the Suitable Law

In a triangle with an SAS configuration, we use the law of cosines first to find the unknown side. The law of cosines is applicable when two sides and the included angle are known.

Key Concepts

triangle configuration

When solving triangle problems, understanding the given configuration is crucial. Each configuration provides certain known sides and angles, defining how we approach the solution. There are five main triangle configurations:

• SAA (Side-Angle-Angle): One side and two angles are known.
• ASA (Angle-Side-Angle): Two angles and the included side are known.
• SSA (Side-Side-Angle): Two sides and a non-included angle are known.
• SAS (Side-Angle-Side): Two sides and the included angle are known.
• SSS (Side-Side-Side): All three sides are known.

Each configuration serves as a blueprint for determining which trigonometric laws to use. For instance, in the original problem, the given sides, \(b\) and \(c\), along with an included angle \(A\), clearly describe a Side-Angle-Side (SAS) configuration. This explicit identification helps streamline the process by choosing the appropriate law to start solving the triangle accurately.

law of sines

The law of sines is vital for solving triangles, especially when specific configurations exist, like ASA, SAA, or SSA. It relates the ratio of the length of a side of a triangle to the sine of its opposite angle. The formula is: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

• When to use: Apply this law when you know either:
  • Two angles and one side (ASA or SAA)
  • Two sides and a non-included angle (SSA)
• Advantages: It provides a straightforward way to find missing lengths or angles in non-right triangles.
• Limitations: Not applicable in SAS or SSS configurations because the angle must be opposite a known side and directly calculable when angles don't lack ambiguity.

For the given problem, even though the law of sines is powerful, the SAS configuration suggests starting with the law of cosines. However, once additional angles are known, the law of sines can help find remaining side lengths or angles.

law of cosines

The law of cosines is an essential tool for solving triangles, particularly useful in scenarios involving SAS or SSS configurations. It connects one side of a triangle to the other two sides and their included angle. The primary formula can be expressed as: \[c^2 = a^2 + b^2 - 2ab \cdot \cos C\] or cyclic permutations for unknown side.

• When to use: Ideal for:
  • Side-Angle-Side (SAS) configurations, where two sides and their included angle are known.
  • Side-Side-Side (SSS) configurations, to find angles if all sides are known.
• Advantages: Allows for computation of a side or angle directly from known measurements, providing a solution pathway for ambiguous angles or where the law of sines is less applicable.

In the original problem with an SAS configuration, the law of cosines shines as it can compute the unknown side directly from two known sides and an angle. Later, knowing all sides could lead to calculating unknown angles with ease, often paving the way for using the law of sines thereafter.