Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$B=72.2^{\circ}, b=78.3 \text { meters, } c=145 \text { meters }$$

Step-by-Step Solution

Answer

No triangle is possible with the given values.

Step 1: Determine the second angle

Using the law of sines, you can find the sine of angle C: \[ \frac{b}{ ext{sin} B} = \frac{c}{ ext{sin} C} \] So, \[ \text{sin} C = \frac{c \times \text{sin} B}{b} = \frac{145 \times \text{sin} (72.2^{\circ})}{78.3} \] Now calculate: \[ \text{sin} B \approx \text{sin}(72.2^{\circ}) \approx 0.9511 \] \[ \text{sin} C = \frac{145 \times 0.9511}{78.3} \approx 1.76 \] Since \(\text{sin} C\) cannot be greater than 1, there is no solution for angle C. Thus, no triangle is possible.

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

No triangle is possible with the given values.

Key Concepts

Triangle Solving

In trigonometry, solving a triangle means finding all of its sides and angles. For any given triangle, we use known values to determine the unknown dimensions. Depending on the given data, different methods like the Law of Sines or the Law of Cosines are utilized. Here, the task was to solve a triangle using the provided information:

• Angle B: 72.2°
• Side b: 78.3 meters
• Side c: 145 meters

With one angle and two sides given, the approach is often to find the remaining angles and the unknown side if needed. Using these measurements, we applied the Law of Sines to search for angle C with the formula:\[\frac{b}{\sin B} = \frac{c}{\sin C}.\]By substituting the known values into this equation, we aimed to solve the problem. This method is useful when two angles and any side or two sides and a non-enclosed angle are known.

Angle Calculation

Angle calculations in a triangle often rely on trigonometric identities like sine, cosine, and tangent. In this particular problem, we attempted to calculate angle C using the Law of Sines. When performing angle calculations, it is crucial to understand the range of trigonometric functions:

• The sine of an angle must be between -1 and 1.
• Angles in a triangle always sum up to 180 degrees.

Following the original exercise, we calculated \(\text{sin} C\) as:\[\text{sin} C = \frac{145 \times \text{sin}(72.2^{\circ})}{78.3} \approx 1.76.\]However, since the sine value exceeded 1, it indicated a problem. A sine value larger than 1 is not possible in real numbers, which meant angle C could not be calculated.This demonstrates an important aspect of trigonometry where you need to verify if calculated trigonometric values fall within valid ranges.

Trigonometry Concepts

The field of trigonometry involves studying the relationships between angles and sides of triangles. One of its key tools is the Law of Sines, especially useful in solving non-right triangles. It is written as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.\]This law allows us to find unknown angles or sides given partial information. Trigonometry also requires careful consideration of functions' properties, such as range and periodicity.When solving triangles, keep in mind:

• Sine values cannot exceed 1 or be negative for a triangle angle.
• Always check for potential ambiguities, such as considering that two triangles could fit the criteria in some cases.
• Use a calculator carefully to ensure precision in complex calculations.

In this problem, identifying that \(\text{sin} C\) was not possible demonstrated important critical thinking in trigonometric analysis.