Application Problem: Law of Sines

Problem Overview

• Scenario: A cell tower is being installed on a hill.
• Goal: Determine the length of a support wire.
• Given:
  • Hill's angle of elevation: 42 degrees.
  • Support wire forms a 15-degree angle with the hill.

Approach: Using the Law of Sines

• Law of Sines Requirement: Need a proportion with only one unknown.
  • Required: at least one angle measure and the length of the opposite side.

Details of the Triangle Setup

• Initial Information:
  • Length of one side given, but not its opposite angle.
  • An angle given, but not the length of its opposite side.

Solving Using Geometry

• Geometry Insight: Use properties of parallel lines.
  • Sketch a segment parallel to the ground.
  • The angle formed by the hill and this segment is 90 degrees.
  • Alternate interior angles are congruent:
    • Hill angle (42 degrees) is congruent to an angle in the triangle.
    • Obtuse angle in the triangle = 90 degrees + 42 degrees = 132 degrees.

Solving for Unknowns

• Known Quantities:

  • Length of the side: 95 meters.
  • Obtuse angle: 132 degrees.

• Find the Acute Angle:

  • Sum of angles in a triangle is 180 degrees.
  • Acute angle = 180 degrees - 132 degrees - 15 degrees = 33 degrees.

Applying the Law of Sines

• Set Up Proportion:

  • ( \frac{\sin(33^\circ)}{95} = \frac{\sin(132^\circ)}{X} )

• Cross-Multiplying to Solve for X:

  • ( X \cdot \sin(33^\circ) = 95 \cdot \sin(132^\circ) )
  • Solve for X:
    • Divide both sides by ( \sin(33^\circ) ):
    • ( X = \frac{95 \cdot \sin(132^\circ)}{\sin(33^\circ)} )

Solution

• Calculate:

  • Ensure calculator is in degree mode.
  • Compute the quotient:
    • ( X \approx 129.6 ) meters

• Conclusion:

  • The length of the support wire should be approximately 129.6 meters.

Summary

• The problem was solved using the Law of Sines and geometric properties of angles formed by parallel lines and transversals.
• Key steps involved identifying known and unknown values, using the sum of interior angles, and applying the Law of Sines to find the required length of the support wire.