Problem 2 - Algebra and Trigonometry

Question

In which of the following cases can we use the Law of Sines to solve a triangle? $$ \mathrm{ASA} \quad \mathrm{SSS} \quad \mathrm{SAS} \quad \mathrm{SSA} $$

Step-by-Step Solution

Answer

The Law of Sines is applicable for ASA and SSA scenarios.

Step 1: Understanding the Law of Sines

The Law of Sines is a relationship between the lengths of the sides of a triangle and the sines of its angles. It is given by the formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). It is primarily used in cases where two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

Step 2: Identify when ASA can use the Law of Sines

In the ASA (Angle-Side-Angle) scenario, we know two angles and the side between them. We can use the Law of Sines because having two angles allows us to find the third angle, and with one side, we can find the remaining two sides.

Step 3: Analyze the SSS case

In the SSS (Side-Side-Side) scenario, we have all three sides, but no angles. The Law of Sines cannot be applied directly here to find angles. Instead, the Law of Cosines is more applicable to find angles when we know all the sides.

Step 4: Consider the SAS case

For the SAS (Side-Angle-Side) case, we know two sides and the included angle. This situation is best solved using the Law of Cosines to find the third side first, not the Law of Sines.

Step 5: Assess the SSA case

In the SSA (Side-Side-Angle) scenario, we know two sides and an angle that is not between them. The Law of Sines can be applied, but caution is needed as this might lead to the ambiguous case where two different triangles could satisfy the given conditions.

Key Concepts

angle-side-angle (ASA)

In the Angle-Side-Angle (ASA) scenario, the Law of Sines becomes quite useful. Here's how it works: when you're given two angles and the side that's nestled comfortably between them, you can easily apply the Law of Sines to find the missing elements of the triangle.
The steps are straightforward:

• First, find the third angle by subtracting the sum of the two known angles from 180°, since the sum of angles in any triangle is always 180°.
• With one side given and now all three angles known, use the Law of Sines formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) to solve for the unknown sides.

This method works perfectly since having two angles ensures that the triangle is well defined, making it a reliable setup for solving using the Law of Sines.

side-side-side (SSS)

The Side-Side-Side (SSS) scenario involves knowing all three sides of a triangle but none of its angles. Unfortunately, the Law of Sines falls short here. It requires at least one angle to get started.
Instead, we turn to the Law of Cosines, which is a powerful tool when dealing with this type of triangle.

• The Law of Cosines allows you to calculate the angles by using the formula: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).
• This formula helps find any one of the angles by rearranging it to solve for \( \cos(C) \), \( \cos(A) \), or \( \cos(B) \).

Once an angle is found, you can then use the Law of Sines if needed to find the remaining angles or verify your results. The SSS case is primarily a job for the Law of Cosines initially.

side-angle-side (SAS)

The Side-Angle-Side (SAS) scenario provides you with two sides and the angle between them. The direct application of the Law of Sines isn't possible initially here.
For this arrangement:

• You should start with the Law of Cosines to determine the third side, with the formula: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).
• Once the third side is found, you can then find the remaining angles if necessary, using the Law of Sines, simplifying the process thereafter.

This method of mixing the Laws ensures that we can tackle a problem from different angles, literally in this geometric challenge, and unlock the complete triangle dimensions using the SAS configuration.

side-side-angle (SSA)

In the Side-Side-Angle (SSA) setup, you are given two sides and one angle that is not enclosed by the sides. The Law of Sines is applicable here, but with caution: it may lead to an ambiguous case.
The triangle might form in more than one way:

• Start by applying the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \), this relation helps to find one of the unknown angles.
• Check for the "ambiguous case"; sometimes, a given SSA can result in two differnt potential triangles. This happens because the sine function can yield two possible angles in a range.

To handle this ambiguity, after calculating the possible angle, verify whether another valid triangle configuration exists by checking if both triangles with different angle solutions comply with the triangle inequality theorem.