Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$B=48.2^{\circ}, a=890 \text { centimeters, } b=697 \text { centimeters }$$

Step-by-Step Solution

Answer

There can be two possible triangles or only one triangle if \( \sin A \leq 1 \). Compute \( \sin A \) to check feasibility.

Step 1: Identify the Given Information

We are given a triangle with angle \( B = 48.2^\circ \), side \( a = 890 \) cm, and side \( b = 697 \) cm. We need to find out the number of possible triangles and their dimensions.

Step 2: Use the Law of Sines

The Law of Sines relates the sides and angles of a triangle. It states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We can use this to find \( \sin A \) as follows: \( \sin A = \frac{a \cdot \sin B}{b} = \frac{890 \cdot \sin 48.2^\circ}{697} \). Calculate \( \sin A \).

Step 3: Determine Possible Values for Angle A

Calculate \( \sin 48.2^\circ \) and plug it into the equation to find \( \sin A = \frac{890 \cdot \sin 48.2^\circ}{697} \). If \( \sin A \leq 1 \), the angle \( A \) can be determined. If \( \sin A > 1 \), no triangle exists.

Step 4: Check for Two Possible Angles

If \( \sin A \leq 1 \), there can be two possible angles for \( A \): \( A = \sin^{-1}(\sin A) \) and \( A = 180^\circ - \sin^{-1}(\sin A) \). Calculate both angles and check if each results in a valid triangle.

Step 5: Calculate the Third Angle and Check Triangle Validity

For each possible angle \( A \), calculate \( C = 180^\circ - A - B \). Check if the angles \( A, B, \) and \( C \) form a valid triangle (all angles should sum to 180°).

Step 6: Solve for Missing Sides if Valid

If a valid triangle is formed, use the Law of Sines to solve for the missing side \( c \): \( \frac{c}{\sin C} = \frac{b}{\sin B} \) or \( \frac{c}{\sin C} = \frac{a}{\sin A} \). Calculate the length for the side \( c \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the lengths and angles of triangles. In triangle solving, specifically, we use trigonometric ratios such as sine, cosine, and tangent to explore these relationships. The Law of Sines, which is applicable here, is a key tool in trigonometry for solving triangles. It states the ratio of the length of a side of a triangle to the sine of its opposite angle is constant across all sides and angles of the triangle. This relationship is fundamental to the problem at hand where knowing two sides and one angle allows us to determine other triangle components.

Triangle Solving

Solving a triangle involves finding the unknown lengths and angles when some of the elements (sides or angles) are known. In our exercise, we are given two sides and one angle. By applying the Law of Sines, we express the unknown angle in terms of known quantities. This step is crucial because it transforms abstract geometric principles into calculable expressions that solve for unknowns. In practical terms, you aim to:

• Find missing angles using known side and angle values.
• Confirm the number of possible triangles based on given data.
• Calculate any remaining side if the triangle is possible.

Remember, a triangle can either have no solution, one solution, or, as in some unique cases, two solutions.

Angle Calculation

Calculating angles accurately is essential for a valid triangle solution. When you calculate \( \sin A \), you first check if it is between -1 and 1, which is a necessary condition for the existence of an angle. If \( \sin A > 1 \), then no such triangle can exist. On the other hand, if \( \sin A \) is within the range, you'll use the inverse sine function (\( \sin^{-1} \)) to find angle A. Since trigonometric functions can be periodic and symmetric, it provides an opportunity for two different angles that could form a triangle leading to the potential of having two distinct solutions for a single problem.

Ambiguous Case

When dealing with the Law of Sines, the ambiguous case arises particularly in SSA (Side-Side-Angle) scenarios. In this situation, two potential triangles can be formed because the initial information allows for two different configurations. For example, when \( \sin A \) is calculated, both \( A = \sin^{-1}(\sin A) \) and \( 180^{\circ} - \sin^{-1}(\sin A) \) need observation.

• Evaluate both angles obtained to ensure they lead to valid triangles.
• Verify that the total angle sum for each calculated configuration does not exceed 180°.

This case needs careful consideration because each configuration provides a potentially valid triangle fitting the given conditions.