Applications of Trigonometry in Real-Life Problems

Overview

Trigonometry plays an essential role in solving real-life problems in various fields like physics, engineering, navigation, and architecture. Its applications extend to modeling periodic phenomena, calculating heights and distances, and analyzing waveforms, among others. In this section, we will explore some of these practical applications.

Modeling Periodic Phenomena

Trigonometric functions, especially sine and cosine, are widely used to model cyclical and periodic phenomena, such as sound waves, light waves, and the motion of pendulums. These functions help us describe quantities that repeat in regular intervals, such as tides, seasonal temperatures, and electrical currents.

Calculating Heights and Distances

Trigonometry is often used to measure heights and distances that are difficult to measure directly. By using indirect methods and trigonometric ratios, we can calculate the height of a building, the distance across a river, or the height of a mountain.

Analyzing Waves

Trigonometric functions such as sine and cosine are also used to analyze waveforms in physics and engineering. Waves, such as sound waves, electromagnetic waves, and seismic waves, are often modeled using sine and cosine functions to understand their properties like amplitude, frequency, and phase.

Key Concepts

  • Cyclical Behavior: Trigonometric functions model phenomena that repeat at regular intervals, such as tides, sound waves, and alternating current.
  • Indirect Measurement: Using trigonometry to calculate distances or heights indirectly, often through angles and known lengths.
  • Wave Analysis: Using trigonometric functions to model and analyze various types of waves, including sound waves, light waves, and seismic waves.

Practice Problems

  1. Using a 30-foot ladder, find the height of a building if the angle of elevation is 60°.
    Solution

    We can use the sine function since we know the length of the ladder (hypotenuse) and the angle of elevation:

    sin(60°) = height / 30

    Solving for height:

    height = 30 * sin(60°) ≈ 30 * 0.866 = 25.98 feet
  2. A sound wave has a frequency of 5 Hz. Using trigonometry, find the period of the wave.
    Solution

    The period is the reciprocal of the frequency:

    Period = 1 / Frequency

    For a frequency of 5 Hz:

    Period = 1 / 5 = 0.2 seconds
  3. A ship is 300 meters from a cliff, and the angle of elevation from the ship to the top of the cliff is 45°. How high is the cliff?
    Solution

    We can use the tangent function to calculate the height of the cliff:

    tan(45°) = height / 300

    Since tan(45°) = 1:

    height = 300 * 1 = 300 meters
  4. A tower casts a shadow of 50 meters when the angle of elevation of the sun is 35°. What is the height of the tower?
    Solution

    We use the tangent function:

    tan(35°) = height / 50

    Solving for height:

    height = 50 * tan(35°) ≈ 50 * 0.7002 = 35.01 meters
  5. A spring oscillates with a maximum displacement of 0.2 meters and a frequency of 10 Hz. Find the amplitude of the oscillation.
    Solution

    The amplitude of an oscillation is the maximum displacement from the equilibrium position. For this problem, the amplitude is 0.2 meters, as given in the problem statement.

  6. Two points on the surface of a lake are 500 meters apart. The angle of depression from one point to the other is 25°. Use trigonometry to find the vertical distance between the two points.
    Solution

    We use the tangent function for this problem:

    tan(25°) = vertical distance / 500

    Solving for the vertical distance:

    vertical distance = 500 * tan(25°) ≈ 500 * 0.4663 = 233.15 meters
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