Problem 34 - Precalculus

Question

The measures of two sides and an angle are given. Determine whether a triangle (or two) exist, and if so, solve the triangle(s). $$b=15.3, c=27.2, \gamma=11.6^{\circ}$$

Step-by-Step Solution

Answer

No valid triangle can be formed with the given measurements.

Step 1: Check if a Triangle Exists (SSA Condition)

We are given two sides and a non-included angle, known as the SSA condition. We need to determine if a triangle exists. Using the Law of Sines, solve for \( a \) using \( \sin A = \frac{a \sin C}{c} \). Begin by calculating \( \sin \gamma \): \( \sin 11.6^\circ \approx 0.2010 \).

Step 2: Calculate Possible Values for Side a

With \( b = 15.3 \) and \( \sin \gamma = 0.2010 \), use the Law of Sines: \[ \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \] which gives: \[ \sin \beta = \frac{b \sin \gamma}{c} = \frac{15.3 \times 0.2010}{27.2} \approx 0.1134 \]. Check \( \sin \beta \) to determine if it can yield a valid angle.

Step 3: Determine the Possible Angles for \( \beta \)

Calculate \( \beta \) using \( \sin^{-1}(\sin \beta) = \sin^{-1}(0.1134) \approx 6.52^\circ \). Since sine has an ambiguous case, check \( 180^\circ - 6.52^\circ \approx 173.48^\circ \). A triangle occurs if the sum of angles (including \( 11.6^\circ \)) is less than \( 180^\circ \).

Step 4: Verify Solutions for Triangles

For \( \beta = 6.52^\circ \), \( \gamma = 11.6^\circ \), solve for \( \alpha = 180^\circ - 6.52^\circ - 11.6^\circ \approx 161.88^\circ \), which is impossible. For \( \beta = 173.48^\circ \), \( \alpha = 180^\circ - 173.48^\circ - 11.6^\circ \approx -5.08^\circ \), also impossible. Thus, neither configuration forms a valid triangle.

Key Concepts

SSA Condition

The SSA condition, meaning Side-Side-Angle, is one of the scenarios in triangle problems where you're given two sides and a non-included angle. It’s like having incomplete information about a triangle. This particular condition can be tricky because it might not always lead to a single triangle.
The key challenge with SSA is determining if it's even possible to form a triangle with the given data. To explore this, we often use the Law of Sines. This helps check the relationships between the sides and angles given. If they satisfy the necessary criteria, a triangle (or two) might exist. But if not, no triangle is possible.
Remember, the SOS condition is notorious for leading into what's known as the 'ambiguous case', which we will explore further.

Triangle Existence

To check if a triangle can exist under the SSA condition, use the Law of Sines. It's a powerful tool because it connects the sides and angles through: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This formula helps us find out if the given sides and angles can form a triangle.
In the given problem, calculate the sine of the angle provided (11.6°) first. Then, we see if using these sides with the Law of Sines gives us a possible angle. If the calculated angle is valid, then a triangle might exist.
If the resulting angles after calculation do not make sense (for instance, if they don't sum up to less than 180°), this means no triangle can form with the given SSA setup.

Angle Calculation

Calculating angles effectively is essential for solving triangles, especially under the SSA condition. Begin with the given side lengths and angle, and apply the Law of Sines. For the problem, you are first solving for the sine of an angle, often leading to two possible angles due to the inverse sine function.
For example, when you find one possible angle (like 6.52° in the example), be aware there's often another, calculated as 180° minus that angle (173.48° in our original problem). This is due to the nature of the sine function, which is positive in both the first and second quadrants.

• Check if adding up the angles gives a valid sum (less than 180°).
• If valid, a triangle or even two triangles might exist.
• If not valid, revisit and see if calculations or conditions were misunderstood.

Ambiguous Case

The ambiguous case in an SSA scenario is intriguing because it allows, sometimes, two possible triangles from the same set of measurements.

Here's where angle calculation becomes tricky. When you get two potential angles from a sine calculation—like 6.52° and 173.48°—it shows this dual possibility. But remember, not every solution results in a practical triangle. Often, after verifying, you may realize both triangles won't truly exist when fitting all components together (like in the earlier exercise).
Always ensure to:

• Check angles and make sure their sum with the known angle does not exceed 180°.
• Verify both possible angles (first and the supplementary) to see real-world plausibility.

This means carefully analyzing each potential triangle setup to confirm if they can genuinely exist.