Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$A=142.13^{\circ}, b=5.432 \text { feet, } a=7.297 \text { feet }$$

Step-by-Step Solution

Answer

The triangle with these parameters exists uniquely with \( B = 14.24^\circ\) and \( C = 23.63^\circ \).

Step 1: Identify Triangle Type

Given angle \(A = 142.13^\circ\), side \(b = 5.432\) feet, and side \(a = 7.297\) feet, we observe that \(A\) is an obtuse angle (greater than \(90^\circ\)). This indicates that triangle is an obtuse triangle.

Step 2: Apply Law of Sines to Find Angle B

We use the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \) to find \( B \). \( \sin B = \frac{b \cdot \sin A}{a} \). Substituting the known values: \( \sin B = \frac{5.432 \cdot \sin 142.13^\circ}{7.297} \).

Step 3: Calculate \(\sin B\) and Find \(B\)

First calculate \(\sin 142.13^\circ\). Then use the Law of Sines equation: \( \sin B = \frac{5.432 \times \sin 142.13^\circ}{7.297} \). Calculate \(B = \arcsin(\sin B)\).

Step 4: Conclusion on Triangle Existence

Since \(\sin B\) must be between \(-1\) and \(1\), check numerical value. If possible, calculate \( B \). Confirm \(B\) is valid and check the third angle \(C = 180^\circ - A - B\). Check for any impossibility due to undefined trigonometric values.

Key Concepts

Law of Sines

The Law of Sines is an important concept in solving triangles, especially oblique triangles, which are not right-angled. This law helps relate the sides and angles of any triangle. When you have a side-angle-side (SAS) or angle-side-angle (ASA) arrangement, the Law of Sines becomes your best friend.

In general, the Law of Sines states:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Where:

• \(a, b, c\) are the lengths of the sides of the triangle.
• \(A, B, C\) are the angles opposite those sides respectively.

To solve a triangle using this law, like in our example where angle \(A\) is known and side \(a\) opposite \(A\) is given, along with another side \(b\), you can set up the equation:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} \)

Then, you can solve for \( \sin B \) by rearranging the formula:

• \( \sin B = \frac{b \cdot \sin A}{a} \)

This process will help you find the other unknown angles in the triangle and continue solving for all components.

Obtuse Angles

Understanding obtuse angles, particularly in the context of triangles, is crucial for solving them accurately. An obtuse angle is one that measures greater than \(90^\circ\) but less than \(180^\circ\).

Obtuse triangles have some unique properties to keep in mind. Only one angle in an obtuse triangle can be obtuse. The sides opposite the obtuse angle are always the longest in the triangle.

In our exercise, the angle \(A\) is \(142.13^\circ\), making it clear we're dealing with an obtuse triangle:

• The angle exceeds \(90^\circ\), confirming its obtuse nature.
• Because of this, the calculations become slightly tricky when using trigonometric identities, primarily when using the Law of Sines or finding other angles.

Keep in mind that the sine of an obtuse angle is positive because sine only considers the vertical component of the angle, which is positive in the second quadrant of the circle. However, solving for other angles becomes straightforward once you have used the Law of Sines formula properly at the beginning.

Triangle Existence

The concept of triangle existence forms the foundation for deciding whether a triangle can be created from given elements like angles and sides. Basic criteria, known as the triangle inequalities, must be satisfied. These specify that:

• The sum of any two sides must be greater than the third side.
• No side can be zero or negative in length.

In addition to these, the sum of internal angles must always equal \(180^\circ\). When solving triangles, especially with obtuse or acute angles, you must ensure your calculations keep these constraints in check.

In our example, even after finding \(B\) using the Law of Sines, we must verify:

• That calculated \(\sin B\) lies within the valid range of \(-1\) to \(1\), ensuring a real angle \(B\) can exist.
• Whether \(A + B + C = 180^\circ\) ensures all angles fit together to form a valid triangle.

If any condition falters, the triangle simply cannot exist with the given measurements. So always double-check these foundational principles when applying trigonometric methods to solve triangles.