Problem 16 - Algebra 2 and Trigonometry

Question

Max has a triangular garden. He measured two sides of the garden and the angle opposite one of these sides. He said that the two sides measured 5 feet and 8 feet and that the angle opposite the 8 -foot side measured 75 degrees. Can a garden exist with these measurements? Could there be two gardens of different shapes with these measurements? Write the angle measures and lengths of the sides of the garden(s) if any.

Step-by-Step Solution

Answer

Yes, one triangle exists with angles of 37.1°, 67.9°, and 75°.

Step 1: Understanding the Problem

We need to determine if a triangle can exist with the given measurements and whether two different triangles can exist with these measurements.

Step 2: Applying the Law of Sines

Given: Side 1 = 5 feet, Side 2 = 8 feet, Angle opposite Side 2 = 75°. We use the Law of Sines to find the angle opposite the 5-foot side. The Law of Sines is: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]Let's find \( \angle A \):\[\frac{5}{\sin A} = \frac{8}{\sin 75°}\] Solving for \( \sin A \), we get:\[\sin A = \frac{5 \times \sin 75°}{8}\]

Step 3: Calculating \( \sin A \)

Calculate \( \sin 75° \) using approximation or a calculator:\[\sin 75° \approx 0.9659\]Thus:\[\sin A = \frac{5 \times 0.9659}{8} = 0.6037\]

Step 4: Finding Possible Values for \( \angle A \)

Since \( \sin A = 0.6037 \), \( A \) can be:- \( \angle A = \sin^{-1}(0.6037) \approx 37.1° \) (Acute angle)- \( \angle A = 180° - 37.1° = 142.9° \) (Obtuse angle)Both angles are possible due to the sine rule in triangles (Ambiguous Case).

Step 5: Finding the Third Angle, \( \angle C \)

For \( \angle A = 37.1° \):\[\angle C = 180° - 75° - 37.1° = 67.9°\]For \( \angle A = 142.9° \):\[\angle C = 180° - 75° - 142.9° = -37.9°\]A negative angle is not feasible in a triangle, so only the acute angle scenario forms a valid triangle.

Step 6: Determining the Results

With \( \angle A = 37.1° \), we have a valid triangle with angles \( 37.1°, 67.9°, \) and \( 75° \). There is no triangle with \( A = 142.9° \) as \( \angle C \) is not valid.

Key Concepts

Ambiguous Case

The ambiguous case arises when solving triangles using the Law of Sines. It specifically occurs when we are given two sides and a non-included angle (SSA Case). The word "ambiguous" means unclear or having more than one interpretation, which can indicate more than one possible triangle that can be formed from the given data.

In Max's garden problem, given sides 5 and 8 feet, and an angle of 75° opposite the 8-foot side, we encounter this case. Using the Law of Sines allows the possibility of two different angle solutions for the angle opposite to the 5-foot side. These are:

• A small acute angle
• A large obtuse angle (since a sine value corresponds to two different angles in 0° to 180° range).

However, only one leads to a valid triangle, thus resolving the ambiguity.

Triangle Existence

For a triangle to exist, certain conditions must be satisfied. Primarily, the sum of the triangle's interior angles must always be 180°. Also, any configuration of sides and angles should adhere to the triangle inequality theorem, i.e., the sum of two sides must be greater than the third side at any given time.

In Max’s case, calculating angles using the Law of Sines shows that when the angle opposite the 5-foot side is acute (approximately 37.1°), all angles neatly total 180° without contradicting any triangle properties. Hence, a valid triangle exists with angles of about 37.1°, 75°, and 67.9°. Conversely, when the angle is obtuse (142.9°), it results in negative angles, which cannot physically exist in a triangle, showing that no second valid triangle can be formed.

Angle Calculation

Angle calculation in triangles often benefits from using trigonometric identities and functions. In scenarios where direct measurement is not possible, the Law of Sines is a reliable tool.

For Max's garden, solving for angle A with given sides and an angle involves setting up the equation from the Law of Sines: \[\frac{5}{\sin A} = \frac{8}{\sin 75°}\] By calculating \(\sin 75°\) with a calculator, we find \(\sin A = 0.6037\). From this sine value, two potential angles can be calculated using inverse sine.

These calculations help ensure each angle and side length works harmoniously to validate the triangle’s existence according to its geometric principles.

Trigonometry Problem Solving

To solve trigonometry problems like Max's garden efficiently:

• Identify known and unknown elements in the triangle setup.
• Choose appropriate trigonometric laws or rules such as the Law of Sines or Cosines.
• Watch out for specific cases like the ambiguous case, which may present multiple solutions.
• Break down complex problems into simpler steps, focusing on finding each missing angle or side sequentially.

For Max’s problem, recognizing the ambiguous case early ensured correct identification of angle solutions via the Law of Sines.

By carefully checking each solution against the fundamental properties of triangles, all calculations guarantee that the resulting configurations align with theoretical expectations.