Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(A=18.75^{\circ}, B=51.53^{\circ}, c=2798\) yards

Step-by-Step Solution

Answer

Side \(a\) is approximately 921.03 yards, side \(b\) is approximately 2226.99 yards, and angle \(C\) is 109.72°.

Step 1: Find Angle C

In any triangle, the sum of the angles is always 180 degrees. To find angle C, we use the formula: \[ C = 180^{\circ} - A - B = 180^{\circ} - 18.75^{\circ} - 51.53^{\circ} \] Calculate: \[ C = 109.72^{\circ} \]

Step 2: Using Law of Sines

To solve the triangle, we apply the Law of Sines, which states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]We'll find side \(a\) first:\[ a = \frac{c \cdot \sin A}{\sin C} = \frac{2798 \cdot \sin 18.75^{\circ}}{\sin 109.72^{\circ}} \]Calculate \(a\):\[ a \approx 921.03 \text{ yards} \]

Step 3: Solve for Side b

Using the same Law of Sines, solve for side \(b\):\[ b = \frac{c \cdot \sin B}{\sin C} = \frac{2798 \cdot \sin 51.53^{\circ}}{\sin 109.72^{\circ}} \]Calculate \(b\):\[ b \approx 2226.99 \text{ yards} \]

Key Concepts

Law of Sines

The Law of Sines is a crucial tool in trigonometry for solving triangles, especially when we are given certain angles and one side. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles in the triangle. This means for any triangle ABC:

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

This law can only be applied to triangles where enough information is given, such as two angles and a side (AAS or ASA conditions), or two sides and a non-included angle (SSA condition). It's particularly useful when you can't use the Pythagorean theorem because the triangle isn't a right triangle. This property allows us to find unknown sides if two angles and one side are known, like in the solution where we were given angles \( A \) and \( B \) and side \( c \). By using this law, we calculated the other two sides \( a \) and \( b \) based on the known sides and the calculated angle C.

Triangle Solving

Solving a triangle means finding all its unknown sides and angles. This involves using given information to determine the remaining characteristics of the triangle. The process might seem complex at first, but it becomes manageable by following a series of systematic steps.

• Identify what you know: Given measures in our problem include angles \( A = 18.75^{\circ} \), \( B = 51.53^{\circ} \), and side \( c = 2798 \) yards.
• Use the Angle Sum Property first, if needed, to find any missing angles.
• Next, apply the Law of Sines to find an unknown side if sufficient angles and a side are known.

This process generally involves a combination of direct computation and the use of geometric relationships, such as those provided by the Law of Sines or, where applicable, the Law of Cosines. Our task was simplified by having two angles and one side given, allowing us to proceed directly to calculate the remaining components of the triangle.

Angle Sum Property

The Angle Sum Property of triangles is a fundamental rule in geometry. It states that the sum of the interior angles of any triangle is always \( 180^{\circ} \). This property is essential when solving triangles because it allows us to find the missing angle if we know the other two.

• The formula used is: \( A + B + C = 180^{\circ} \)
• If two angles are known, the third can be found by rearranging to \( C = 180^{\circ} - A - B \)

In our example, with angles \( A = 18.75^{\circ} \) and \( B = 51.53^{\circ} \), we find angle \( C \) by plugging these values into our formula, which gives \( C = 109.72^{\circ} \). This method is straightforward and does not depend on any other elements of the triangle, making it a reliable first step in solving for unknowns.