Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. There may be two, one, or no such triangle. $$C=52.3^{\circ}, a=32.5 \text { yards, } c=59.8 \text { yards }$$

Step-by-Step Solution

Answer

The triangle is valid with angles \(A \approx 25.4^{\circ}\), \(B \approx 102.3^{\circ}\), and side \(b \approx 73.9\) yards.

Step 1: Analyze the Given Data

We have a triangle with angle \(C = 52.3^{\circ}\) and sides \(a = 32.5\) yards and \(c = 59.8\) yards. We need to decide if the given triangle scenario is possible, and if so, calculate any unknown angles or sides.

Step 2: Check Triangle Validity Using the Sine Rule

We will use the Law of Sines to check for triangle possibilities. According to the Law of Sines, \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Substituting the given values, \( \frac{32.5}{\sin A} = \frac{59.8}{\sin 52.3^{\circ}} \). Solve for \( \sin A \), yielding \( \sin A = \frac{32.5 \cdot \sin 52.3^{\circ}}{59.8} \). Calculate to check if \( \sin A \) is within the range [0,1].

Step 3: Calculate \(\sin A\)

Calculate \(\sin C = \sin 52.3^{\circ} \approx 0.7880 \). Then, \( \sin A = \frac{32.5 \cdot 0.7880}{59.8} \approx 0.4282 \). Since \( \sin A \) is within [0,1], angle \(A\) is possible. Calculate \(A\) using \(A = \arcsin(0.4282)\) (principal value).

Step 4: Determine Angle \(A\)

Using \(A = \arcsin(0.4282)\), calculate \(A \approx 25.4^{\circ}\). Since there is a possibility of an ambiguous case, check if an additional angle is possible by computing \(A' = 180^{\circ} - 25.4^{\circ} = 154.6^{\circ}\). However, since \(C = 52.3^{\circ}\), \( A' \) would make the sum of angles greater than 180°, which isn't possible. So \(A \approx 25.4^{\circ}\) is the valid angle.

Step 5: Compute Angle \(B\) from Sum of Angles

Calculate \(B = 180^{\circ} - A - C = 180^{\circ} - 25.4^{\circ} - 52.3^{\circ} \approx 102.3^{\circ}\).

Step 6: Use the Law of Sines to Find Side \(b\)

Use the Law of Sines: \( \frac{b}{\sin B} = \frac{a}{\sin A} \). Substituting the known values gives \( \frac{b}{\sin 102.3^{\circ}} = \frac{32.5}{\sin 25.4^{\circ}} \). Calculate \(b\) by rearranging: \( b = \frac{32.5 \cdot \sin 102.3^{\circ}}{\sin 25.4^{\circ}}\).

Step 7: Solve for Side \(b\)

Calculate \(\sin 102.3^{\circ} \approx 0.9744\) and \(\sin 25.4^{\circ} \approx 0.4282\). Substitute into the equation for \(b\): \( b = \frac{32.5 \times 0.9744}{0.4282} \approx 73.9\) yards.

Key Concepts

Law of Sines

The Law of Sines is an essential principle in trigonometry used to solve triangles. This law relates the ratios of each side of a triangle to the sine of its respective opposite angle. The formula is given as:

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

This equation is helpful for finding missing angles or sides of a triangle, particularly when dealing with non-right triangles. It's crucial to have at least one angle-side opposite pair known. To see it in action, if side \( a \) and angle \( A \) are known, and \( c \) and \( C \) are also available, one can find either the angle \( A \) or side \( b \) by setting up the proper ratio. The Law of Sines plays a critical role in solving the provided exercise since it is pivotal in checking the validity of a triangle and determining unknown sides and angles.

Triangle Solving

Solving a triangle generally means determining all unknown sides and angles of a triangle when certain elements such as specific sides and angles are known. The process depends on the available information:

• Two angles and one side (AAS or ASA method)
• Two sides and an angle not between them (SSA method)
• Three sides (SSS method)
• Two sides and the angle between them (SAS method)

The exercise at hand requires solving a triangle using the SSA (Side-Side-Angle) method. This method, however, can sometimes result in two possible triangles, hence an ambiguous case might occur. Understanding how to apply the Law of Sines is vital when solving triangles through any method that involves angles as it helps relate angles to their respective sides, helping clarify which scenario is physically possible.

Ambiguous Case

The ambiguous case arises specifically when using the SSA condition (two sides and a non-included angle) and refers to situations where two different triangles could potentially satisfy the same given conditions.

• One triangle solution
• Two triangle solution
• No triangle solution

The ambiguity happens because the known angle and the opposite side might suggest two possible triangles due to the nature of the sine function, which has an identical sine value for angles that are supplementary (i.e., add up to 180°). For example, if angle \( A \) can be a specific value derived by \( \arcsin \), its supplementary angle \( 180° - A \) might also be valid. However, checking the sum of the calculated angles is key. If adding extra angles leads to a sum that exceeds 180°, the alternative scenario is not possible, just like in this exercise where an alternative angle \( A' \) was derived but invalidated due to angle constraints.

Angle Calculation

Calculating an angle in a triangle means finding its measure using known sides and angles, often utilizing trigonometric principles like the Law of Sines.In a triangle with one angle and its opposite side known, alongside another non-included side, it involves:

• Determining \( \sin A \) using \( \sin A = \frac{a \cdot \sin(C)}{c} \)
• Computing the angle using arcsine: \( A = \arcsin(\sin A) \)

The arcsine function helps find the principal angle value from a calculated sine ratio. An understanding of sine range \([0,1]\) ensures the calculated \( \sin A \) implies a realizable angle. Supplementary angles should be checked to account for any ambiguous case possibilities. Finally, triangle angle sum rule, \( A + B + C = 180° \), is used to finalize any unknown angle, ensuring all calculated angles fit within triangular constraints. This method results in a complete understanding of the triangle's dimensions, as demonstrated in this solution, where calculating angle \( A \) helped find the rest of the triangle's angles and sides.