Explain how the Law of Sines can be used to solve for an unknown side in a triangle given two angles and one side.

The Law of Sines is a powerful tool in trigonometry that allows you to find unknown sides or angles in any triangle, provided certain information is known. Specifically, when you are given two angles and one side of a triangle, the Law of Sines can be used to determine the lengths of the unknown sides. Here's a step-by-step explanation of how to apply the Law of Sines in such a scenario. ## **Understanding the Law of Sines** The Law of Sines states that in any triangle (not necessarily a right triangle), the ratios of the lengths of a side to the sine of its opposite angle are equal. Mathematically, this is expressed as: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] Where: - \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle. - \( A \), \( B \), and \( C \) are the measures of the angles opposite those sides, respectively. ## **Steps to Find an Unknown Side** Given: - Two angles: Let's denote them as \( A \) and \( B \). - One side: Let's denote the known side as \( a \), opposite angle \( A \). **Step 1: Find the Third Angle** Since the sum of the internal angles in any triangle is \( 180^\circ \), you can find the third angle \( C \) as follows: \[ C = 180^\circ - A - B \] **Step 2: Apply the Law of Sines** Now, you can set up the proportion using the Law of Sines. Suppose you want to find side \( b \) opposite angle \( B \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] **Step 3: Solve for the Unknown Side** Rearrange the equation to solve for \( b \): \[ b = \frac{a \cdot \sin B}{\sin A} \] Similarly, if you need to find side \( c \) opposite angle \( C \): \[ c = \frac{a \cdot \sin C}{\sin A} \] ## **Example Problem** **Given:** - Angle \( A = 30^\circ \) - Angle \( B = 45^\circ \) - Side \( a = 10 \) units (opposite angle \( A \)) **Find:** - Lengths of sides \( b \) and \( c \) **Solution:** **Step 1: Find the Third Angle \( C \)** \[ C = 180^\circ - 30^\circ - 45^\circ = 105^\circ \] **Step 2: Apply the Law of Sines to Find Side \( b \)** \[ \frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ} \] \[ \frac{10}{0.5} = \frac{b}{0.7071} \] \[ 20 = \frac{b}{0.7071} \] \[ b = 20 \times 0.7071 \approx 14.14 \text{ units} \] **Step 3: Apply the Law of Sines to Find Side \( c \)** \[ \frac{10}{\sin 30^\circ} = \frac{c}{\sin 105^\circ} \] \[ \frac{10}{0.5} = \frac{c}{0.9659} \] \[ 20 = \frac{c}{0.9659} \] \[ c = 20 \times 0.9659 \approx 19.32 \text{ units} \] **Conclusion:** - Side \( b \approx 14.14 \) units - Side \( c \approx 19.32 \) units ## **Key Points to Remember** 1. **Sum of Angles:** Always ensure that the sum of the given angles does not exceed \( 180^\circ \). If it does, there is no such triangle. 2. **Unique Solution:** When two angles and one side are given (known as the Angle-Angle-Side or AAS condition), the triangle is uniquely determined. There's no ambiguity, unlike some cases in the Side-Side-Angle (SSA) scenario. 3. **Calculators in Degrees or Radians:** Ensure that your calculator is set to the correct mode (degrees or radians) based on the units of the angles provided. 4. **Accuracy:** When performing calculations involving sine values, using more decimal places can increase accuracy, especially for more precise applications. By following these steps and keeping these key points in mind, you can effectively use the Law of Sines to solve for unknown sides in a triangle when given two angles and one side.