Problem 67 - Precalculus

Question

Explain the mistake that is made. Solve the triangle \(b=3, c=4,\) and \(\alpha=30^{\circ}\) Solution: Step 1: Find \(a\) Apply the Law of Cosines. \(a^{2}=b^{2}+c^{2}-2 b c \cos \alpha\) Let \(b=3\) \(c=4,\) and \(\alpha=30^{\circ} . \quad a^{2}=3^{2}+4^{2}-2(3)(4) \cos 30^{\circ}\) Solve for \(a, \quad a \approx 2.1\) Step 2: Find \(\gamma\) \(\begin{array}{ll}\text { Apply the Law } & \frac{\sin \alpha}{a}=\frac{\sin \gamma}{c} \\ \text { of sines. }\end{array}\) Solve for \(\sin \gamma\) \(\sin \gamma=\frac{c \sin \alpha}{a}\) Solve for \(\gamma\) \(\gamma=\sin ^{-1}\left(\frac{c \sin \alpha}{a}\right)\) Let \(a=2.1, c=4\) and \(\alpha=30^{\circ} . \quad \gamma \approx 72^{\circ}\) Step 3: Find \(\beta\) \(\alpha+\beta+\gamma=180^{\circ} \quad 30^{\circ}+\beta+72^{\circ}=180^{\circ}\) Solve for \(\beta\) \(\beta \approx 78^{\circ}\) \(a \approx 2.1, b=3, c=4, \alpha=30^{\circ}, \beta \approx 78^{\circ},\) and \(\gamma \approx 72^{\circ}\) This is incorrect. The longest side is not opposite the largest angle. What mistake was made?

Step-by-Step Solution

Answer

The mistake was in the calculation of side \(a\) using the Law of Cosines, which led to a misalignment with angle sizes and their respective opposing sides.

Step 1: Misuse of Law of Cosines

The given solution for finding side \(a\) using the Law of Cosines incorrectly used the values. The formula \(a^2 = b^2 + c^2 - 2bc \cos \alpha\) was applied, with \(b=3\), \(c=4\), and \(\alpha = 30^{\circ}\). Calculating: \(a^2 = 3^2 + 4^2 - 2(3)(4) \cos 30^{\circ}\). The mistake is found in the final calculation leading to \(a \approx 2.1\), which seems miscalculated, as this leads to an incorrect triangle configuration where the longest side should be opposite the largest angle, but \(a < b < c\).

Step 2: Re-evaluating Calculation

Recognizing the error, recalculate the Law of Cosines step accurately:\[a^2 = 9 + 16 - 24(\sqrt{3}/2),a^2 = 25 - 12\sqrt{3},a \approx 4.96.\]Rather than the previously computed value, this correction reflects more logically in the subsequent steps, aligning angle-size with side-length.

Step 3: Correct Order of Angles and Sides

Given the correct side lengths, identify the angles using proper calculation. The result of the Law of Cosines was incorrect in reflecting the size of angle \(\gamma\) versus \(\beta\). With side \(a\) larger than previously stated, subsequently figuring angle \(\beta\) will exhibit larger than \(\gamma\), based on their correct placement opposite the smallest and presumed middle side respectively.

Step 4: Conclusion: Recalculating Angles Adjusted

Reapply the Law of Sines correctly to follow the correction. With angle \(\alpha = 30^{\circ}\) and utilizing correct \(a\), solve for \(\gamma\) and \(\beta\) anew. Accurate recalculation should find imbalance corrected: \(\gamma\) remains smallest, adjusting \(\beta\) to be largest. \(a\) is associated with a significant angle size controversially reduced earlier.

Key Concepts

Triangle Solution

Solving a triangle involves finding all the unknown sides and angles of a given triangle. For this particular problem, we were given two sides, b and c, and one angle, \(\alpha\), which makes it a SSA (Side-Side-Angle) situation.
The main tools to solve such a triangle are the Law of Cosines and Law of Sines.

• The Law of Cosines is typically used to find a side when two sides and one angle are known.
• The Law of Sines is beneficial to determine an angle, especially in ambiguous cases.

To effectively solve the triangle, ensure the side-length and angle correlations align, following rules of trigonometry. Pay attention to checking that the longest side is opposite the largest angle, which helps confirm calculations are accurate.

Angle-Side Relationship

The angle-side relationship in a triangle is fundamental for understanding its geometric properties. In any triangle, the larger angle is always opposite the longer side, and the smaller angle opposite the shorter side.
Misaligned calculations can lead to errors, such as in this exercise. Initially, we inaccurately found the longest side not opposite the largest angle, contradicting this principle.

• For a correct triangle, if side \(a\) is computed to be longer, then the angle opposite it should also reflect greater than smaller angles.
• Ensure that every adjustment in one measurement reflects logically in other measurements.

This understanding of angle and side relations helps maintain the internal consistency of a triangle's measurements, critical for accurate problem solving.

Law of Sines

The Law of Sines is a powerful tool in solving triangles, particularly useful when dealing with angles. This law states: \[ \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c} \]
It allows us to find unknown angles or sides when knowing specific combinations. In cases like the provided example, this law helps find unknown angles.

• When using the Law of Sines, care must be taken, especially in ambiguous cases (SSA configuration), to ensure the calculated angles are reasonable and follow the triangle's logical structure.
• Proper recalibration or verification is necessary to confirm that derived angles fit within the triangle's total 180° constraints.

When one angle is known, you can use the law to confirm other angles, provided no contradictions arise. Such validations help rectify any erroneous calculations from a prior misstep.

Precalculus Problem-Solving

Precalculus involves solving problems, like triangles, using foundational algebra and trigonometry principles. This lays the groundwork for more advanced mathematics. Break down a problem by identifying known quantities and deploying relevant mathematical laws judiciously.
The initial exercise contains straightforward missteps emphasizing two precalculus problem-solving strategies:

• Double-check computational steps and derived values to prevent errors cascading through calculations.
  
• Apply logical reasoning to ensure outcomes (like side-angle relations) match expected geometric properties.
  
• Recognize problems of alignment, like sides versus opposite angles, as indicators something might be amiss.
  
• Acknowledge tools like the Law of Cosines and Sines as not just computational aids but verification tools.

Ensuring accuracy in precalculus builds foundational confidence for upper-level courses, centralizing reliability in all computations.