Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(A=37^{\circ}, C=95^{\circ}, c=18\) meters

Step-by-Step Solution

Answer

The angles are A = 37°, B = 48°, C = 95°. The sides are a ≈ 11.41 m, b ≈ 13.36 m, c = 18 m.

Step 1: Calculate Angle B

To find the third angle of the triangle, use the fact that the sum of angles in a triangle is always 180 degrees. Thus, \(B = 180^{\circ} - 37^{\circ} - 95^{\circ}\). Calculate \(B\).

Step 2: Apply the Law of Sines to Find Side a

The Law of Sines states \(\frac{a}{\sin A} = \frac{c}{\sin C}\). You can rearrange this to solve for \(a\), giving \(a = \frac{c \cdot \sin A}{\sin C}\). Substitute \(A = 37^{\circ}, C = 95^{\circ}, c = 18\) meters, and compute \(a\).

Step 3: Apply the Law of Sines to Find Side b

Similarly, using the Law of Sines \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Solve for \(b\) using the rearrangement \(b = \frac{c \cdot \sin B}{\sin C}\). Substitute the value for \(B\) calculated in Step 1, and compute \(b\).

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in trigonometry, especially helpful when solving triangles. This law relates the sides of a triangle to the sines of its angles. Specifically, if you have a triangle with sides and angles labeled, you can express the Law of Sines with the equation:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In this equation, \(a\), \(b\), and \(c\) represent the lengths of the sides opposite the angles \(A\), \(B\), and \(C\) respectively. The essence of this law is its ability to solve triangles when certain angles and sides are known.

For example, if two angles and one side are known, like in our problem, the Law of Sines helps find the remaining sides and angles. It simplifies calculations by transforming complex trigonometric relationships into straightforward ratios. The law is based on the properties of a circle and is validated by the fact that the sum of angles in a triangle is always 180 degrees. This consistency makes it a reliable method for solving various triangle problems.

Angle Calculations

Calculating the angles in a triangle is often the first step in solving triangle-related problems. This process typically involves using the sum of the angles property, which asserts that the sum of all interior angles in a triangle must equal 180 degrees. This can be represented as:\[A + B + C = 180^{\circ}\]In our example, two angles \(A = 37^{\circ}\) and \(C = 95^{\circ}\) are given. To find the third angle \(B\), we subtract the sum of the known angles from 180 degrees:
\[ B = 180^{\circ} - A - C \]

This straightforward subtraction gives us the value of \(B\), completing the angle calculations for the triangle. Understanding this procedure is foundational because accurate angle measurements are necessary for correctly applying further trigonometric laws, such as the Law of Sines. It enables you to consistently progress to solving for side lengths once the angles are known.

Triangle Solving

Triangle solving involves finding all unknown sides and angles of a triangle using given information. In trigonometry, this typically means using both calculated and given data to completely determine the triangle's properties.

In our example, once we've calculated all angles using the angle sum property, the next step involves using the Law of Sines to find the unknown sides, \(a\) and \(b\).

• Finding side \(a\): Using the formula derived from the Law of Sines: \(a = \frac{c \cdot \sin A}{\sin C}\). You use the known values of \(A\), \(C\), and \(c\) to solve for \(a\).
• Finding side \(b\): Similarly, apply the rearranged Law of Sines: \(b = \frac{c \cdot \sin B}{\sin C}\). Here, you replace the computed angle \(B\) into the equation to find \(b\).

Through this systematic approach, utilizing trigonometric relationships, one can solve any triangle given sufficient initial information. The goal is to derive the complete set of side lengths and angle measures, ensuring a comprehensive understanding of the triangle's geometric properties.