Problem 28 - Precalculus

Question

Solve each triangle. $$\alpha=25^{\circ}, a=6, c=9$$

Step-by-Step Solution

Answer

The remaining sides and angles are \( b \approx 12.87 \), \( \beta \approx 115.77^{\circ} \), and \( \gamma \approx 39.23^{\circ} \).

Step 1: Identify Triangle Type

Given that we have one angle, \( \alpha = 25^{\circ} \), and two sides, \( a = 6 \) and \( c = 9 \), we recognize this as a triangle requiring the use of the Law of Sines and potentially the Law of Cosines.

Step 2: Use the Law of Cosines

First, use the Law of Cosines to find the missing angle \( \gamma \). \[ c^2 = a^2 + b^2 - 2ab \cos(\gamma) \].Given, \( c = 9 \) and \( a = 6 \), substitute to get:\[ 9^2 = 6^2 + b^2 - 2 \times 6 \times b \cos(25^{\circ}) \], \[ 81 = 36 + b^2 - 12b \cos(25^{\circ}) \].We can't proceed with just one angle and side, we're missing another relationship. Instead, let's find another angle using the Law of Sines.

Step 3: Use the Law of Sines

Now apply the Law of Sines to find the angle \( \beta \):\[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \],\[ \frac{6}{\sin(25^{\circ})} = \frac{9}{\sin(\gamma)} \].Solve for \( \sin(\gamma) \):\[ \sin(\gamma) = \frac{9 \times \sin(25^{\circ})}{6} \].Calculate \( \sin(25^{\circ}) \approx 0.4226 \),\[ \sin(\gamma) = \frac{9 \times 0.4226}{6} = 0.6339 \].Find \( \gamma \) using the arcsin function,\( \gamma \approx \arcsin(0.6339) \approx 39.23^{\circ} \).

Step 4: Determine Third Angle

Use the fact that the sum of angles in a triangle is \( 180^{\circ} \) to find \( \beta \):\[ \beta = 180^{\circ} - \alpha - \gamma \],\[ \beta = 180^{\circ} - 25^{\circ} - 39.23^{\circ} \],\( \beta \approx 115.77^{\circ} \).

Step 5: Calculate Third Side

Use the Law of Sines to find \( b \):\[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \].Substituting known values:\[ \frac{6}{\sin(25^{\circ})} = \frac{b}{\sin(115.77^{\circ})} \],\( \sin(115.77^{\circ}) \approx 0.9063 \),\[ b = \frac{6 \times 0.9063}{0.4226} \approx 12.873 \].

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry used to solve triangles. This law works by relating the sides of a triangle to the sines of its angles. This is particularly useful when dealing with non-right triangles where we know:

• Two angles and one side (AAS or ASA configuration), or
• Two sides and an angle opposite one of them (SSA configuration).

The formula is expressed as \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]Here, \( a \), \( b \), and \( c \) are the lengths of the sides, and \( A \), \( B \), and \( C \) are the respective opposite angles.To apply this when solving triangles, identify which parts of your triangle are known. Then, rearrange the formula to solve for unknown parts. For instance, when given angles \(\alpha\) and side \(a\) against another side \(c\), solve for the angle \(\gamma\) using:\[ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \]Once one of the angles is discovered, finding the others becomes a simple application of angle sum properties or reusing the Law of Sines.

Law of Cosines

The Law of Cosines is invaluable when you have triangles where the Law of Sines isn't enough, especially when knowing:

• All three sides (SSS configuration), or
• Two sides and the included angle (SAS configuration).

This law extends the Pythagorean theorem to any triangle, providing the relation:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]This equation connects one angle's cosine with all three side lengths, allowing us to solve for unknowns in the triangle.In practice, if you know two sides \(a\) and \(c\), and one angle \(\alpha\), but not the third side or other angles, start by using the Law of Cosines to find a missing angle:\[ \gamma = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \]After deducing \(\gamma\), employ either the angle sum or the Law of Sines to find remaining unknowns.Thus, the Law of Cosines is both a reliable and robust tool for calculating parts of a triangle that might not initially seem reachable.

Triangle Solving

Solving a triangle involves finding all unknown angles and side lengths. This requires understanding which tools - the Law of Sines or Cosines - are appropriate based on your given information. Here's how you can proceed step-by-step:**1. Identify Given Elements:**

• Determine whether the known quantities pertain to angle-side-angle (ASA), side-angle-side (SAS), side-side-angle (SSA), or side-side-side (SSS).

**2. Choose the Approach:**

• If angles and non-included sides are known, start with the Law of Sines.
• If two sides and an included angle or all sides are known, opt for the Law of Cosines first.

**3. Solve for Unknown Angles or Sides:**

• Calculate unknown angles using angle sum properties: \(\alpha + \beta + \gamma = 180^\circ\).
• Leverage calculated angles to use alongside the Law of Sines or Cosines for missing side lengths.

By following these steps in a structured manner, efficiently solve any triangle and build a deeper understanding of trigonometry principles. Always double-check calculations to minimize errors and ensure all angles and sides correspond correctly.