Problem 32 - Algebra 2

Question

Mateo and Amy are deciding which method, the Law of Sines or the Law of Cosines, should be used first to solve \(\triangle A B C\) . Mateo Begin by using the Law of Sines, since you are given two sides and an angle opposite one of them. Amy Begin by using the Law of Cosines, since you are given two sides and their included angle. Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

Amy is correct if the angle is included; Mateo is correct if the angle is opposite a side.

Step 1: Identify Given Information

In the triangle \( \triangle ABC \), you are given two sides and one angle. You need to determine whether the angle is opposite one of the sides, or if it is the included angle (the one between the two sides). This will guide the choice between using the Law of Sines or the Law of Cosines.

Step 2: Determine the Angle Position

If the angle given is opposite one of the two known sides, you can use the Law of Sines. If the angle is the included angle (between the two given sides), the Law of Cosines should be used.

Step 3: Analyze Mateo's Suggestion

Mateo suggests using the Law of Sines first because he assumes the given angle is opposite one of the sides. The Law of Sines requires a known ratio of side and its opposite angle, so this approach is valid if the angle is indeed opposite a known side.

Step 4: Analyze Amy's Suggestion

Amy suggests using the Law of Cosines, which is appropriate when you are given two sides and the angle between them (the included angle). This method is useful for finding the third side or an unknown angle when the included angle is given.

Step 5: Evaluate Who is Correct

Both suggestions depend on the specific position of the given angle. Mateo is correct if the angle is opposite a known side, allowing for the use of the Law of Sines. Amy is correct if the angle is between the two known sides, making the Law of Cosines the appropriate choice.

Key Concepts

triangle problem solving

Solving triangle problems can initially seem daunting, but by understanding key principles, you can easily decide which strategy to adopt. When faced with a triangle where you know two sides and one angle, the problem becomes a bit of a puzzle. Your first step is to identify if the given angle is opposite to one of the sides or if it’s the included angle, which lies between the two sides.

Here's how it works:

• If the angle is opposite one of the known sides, you can utilize the Law of Sines.
• If the angle is between the two given sides, the Law of Cosines is your best bet.

Successfully solving these types of problems requires assessing what you know and applying the correct mathematical law, either the Law of Sines or Cosines, to find the missing information. Always begin by examining the position of the angle in relation to the sides provided.

Law of Sines

The Law of Sines is a powerful tool when you have a triangle with known values that fit certain conditions. Particularly, it’s beneficial if you know:

• Two angles and one side (AAS or ASA configurations).
• Two sides and a non-included angle (SSA configuration).

This law relates the sides of the triangle to its angles: equation: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]In this equation:

• \(a, b, \) and \(c\) are the lengths of the sides opposite to angles \(A, B, \) and \(C\) respectively.
• \(\sin A, \sin B, \) and \(\sin C\) are the sines of the respective angles.

The Law of Sines can solve a triangle when you have a side-angle pair. It allows you to find unknown angles or sides, depending on the information you have. Keep in mind that if the given angle is opposite a known side, this approach is ideal.

angle-side-angle relationship

Understanding the angle-side-angle (ASA) relationship in triangles can help in choosing the right methodology for solving problems. This relationship occurs when two angles and the side between them are known. With this configuration, the Law of Sines is particularly useful because you can determine the missing side or angle using known information.

In scenario of having two sides and the included angle (commonly known as the Side-Angle-Side or SAS scenario), the Law of Cosines is the appropriate choice, as it allows calculation of a third side or unknown angles. This is because the included angle directly affects the length of the opposite side based on how large or small it is relative to the other sides.

• ASA: Law of Sines is effective.
• SAS: Law of Cosines is effective.

This understanding of angle-side relationships allows for effective problem-solving, leaving no room for confusion when deciding which law to apply to find missing parts of a triangle.