How to Apply Trigonometry: Practical Uses and Insights into Engineering and Astronomy

• Ancient astronomers used trigonometry to calculate distances between stars and planets.
• Today, it aids in determining the position and path of celestial bodies.

• The design of buildings, bridges, and monuments often involves trigonometry to ensure stability and aesthetics.
• It aids architects in creating 3D models on computer software.

• Sailors and pilots have historically used trigonometry to find their direction and distance from a particular landmark.
• Modern GPS systems still employ trigonometric algorithms to provide accurate location data.

• Trigonometry is crucial in understanding concepts like waves, optics, and motion.
• Engineers use it to analyze forces, design machinery, and optimize energy use.

Music:

• Sound waves can be broken down into trigonometric functions.
• Engineers design speakers and instruments considering these wave patterns.

Medicine:

• Imaging techniques like MRI and ultrasound use trigonometric calculations to generate images of the body’s internal structures.

Video Games and Graphics:

• Game developers use trigonometry to simulate realistic movements, shadows, and trajectories.
• It aids in rendering 3D graphics and virtual reality simulations.

Examples

Example 1:
A pilot flying at an altitude of \(6000\) meters sees a landmark at a depression angle of \(10^\circ\). How far is the plane from the landmark on the ground?

Solution:
Using the tangent function:
\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
Here, the opposite side is the altitude of the plane, and the adjacent side is the distance from the landmark.
\( \tan(10^\circ) = \frac{6000}{\text{distance}} \)
\( \text{distance} = \frac{6000}{\tan(10^\circ)} \)
\( \text{distance} \approx 34033 \text{ meters} \)

Example 2:
An architect designs a ramp for a wheelchair with a \(5^\circ\) angle of elevation. If the vertical height of the ramp is \(0.5\) meters, how long is the ramp?

Solution:
Using the sine function:
\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
Here, the opposite side is the height of the ramp, and the hypotenuse is the length of the ramp.
\( \sin(5^\circ) = \frac{0.5}{\text{length}} \)
\( \text{length} = \frac{0.5}{\sin(5^\circ)} \)
\( \text{length} \approx 5.73 \text{ meters} \)

Practice Questions:

1. A lighthouse casts a shadow of \(25\) meters when the angle of elevation of the sun is \(45^\circ\). Calculate the height of the lighthouse.
2. In an amusement park ride, a pendulum swings at an angle of \(30^\circ\) from the vertical. If the length of the pendulum is \(20\) meters, how far above the ground is the pendulum’s bob?

Answers:

1. Approximately \(25\) meters.
2. Approximately \(17.32\) meters above the ground.