What is the Use of Trigonometry in Navigation? | Class 10

S6-SA2-0382

What is the Use of Trigonometry in Navigation Routes?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Trigonometry helps us find positions and directions using angles and distances, which is super important for navigation. It allows pilots, sailors, and even your GPS to figure out 'where am I?' and 'where do I need to go?' by working with triangles formed by different locations.

Simple Example

Quick Example

Imagine you are flying a kite. You know how long the string is (say, 50 meters) and the angle the string makes with the ground (say, 60 degrees). Using trigonometry (specifically, the sine function), you can calculate exactly how high your kite is above the ground, even without climbing up to measure it!

Worked Example

Step-by-Step

Let's say a ship needs to travel from Point A to Point B. Point B is 100 km North and 75 km East of Point A. What is the straight-line distance and the bearing (angle) the ship needs to follow?

1. Draw a right-angled triangle. Point A is at the origin. Move 100 km North (up the y-axis) and 75 km East (along the x-axis) to reach Point B.
---2. The North distance is one leg (opposite side to the angle from East), and the East distance is the other leg (adjacent side).
---3. To find the straight-line distance (hypotenuse), use the Pythagorean theorem: Distance = sqrt((North_distance)^2 + (East_distance)^2).
---4. Distance = sqrt((100)^2 + (75)^2) = sqrt(10000 + 5625) = sqrt(15625) = 125 km.
---5. To find the bearing (angle from North towards East), we can use the tangent function: tan(angle) = (East_distance) / (North_distance).
---6. tan(angle) = 75 / 100 = 0.75.
---7. angle = arctan(0.75) approximately 36.87 degrees.
---8. So, the ship needs to travel 125 km at a bearing of 36.87 degrees East of North.

Answer: The straight-line distance is 125 km, and the bearing is approximately 36.87 degrees East of North.

Why It Matters

Trigonometry is vital for engineers designing bridges, scientists tracking satellites, and doctors using imaging techniques. It helps pilots navigate planes safely, ensuring they reach their destination accurately. Careers in space technology, defense, and even app development for navigation rely heavily on these principles.

Common Mistakes

MISTAKE: Confusing sine, cosine, and tangent ratios, especially which side is opposite or adjacent. | CORRECTION: Always label the sides (Hypotenuse, Opposite, Adjacent) relative to the angle you are working with before applying the SOH CAH TOA rule.

MISTAKE: Using angles in degrees when the calculator is set to radians, or vice versa. | CORRECTION: Double-check your calculator's mode (DEG or RAD) before performing calculations, especially when dealing with inverse trigonometric functions.

MISTAKE: Assuming all navigation problems involve right-angled triangles. | CORRECTION: While many initial problems do, real-world navigation often uses the Law of Sines and Law of Cosines for non-right-angled triangles. Always identify the type of triangle first.

Practice Questions

Try It Yourself

QUESTION: A lighthouse keeper sees a ship at an angle of depression of 30 degrees. If the lighthouse is 60 meters tall, how far is the ship from the base of the lighthouse? | ANSWER: Approximately 103.92 meters

QUESTION: A drone flies 8 km North and then 6 km East. What is its direct distance from the starting point and what is the angle (bearing) it needs to follow from North towards East to return directly? | ANSWER: Distance = 10 km, Angle = approximately 36.87 degrees East of North

QUESTION: Two ships leave a port at the same time. Ship A travels 40 km/hr due East. Ship B travels 30 km/hr at a bearing of 60 degrees North of East. How far apart are the two ships after 2 hours? (Hint: Use Law of Cosines) | ANSWER: Approximately 60.83 km

MCQ

Quick Quiz

Which trigonometric ratio is most directly used to find the height of an object if you know the distance from its base and the angle of elevation?

Cosine

Sine

Tangent

Secant

The Correct Answer Is:

Tangent relates the opposite side (height) to the adjacent side (distance from base) for a given angle of elevation, making it ideal for this scenario. Sine relates opposite to hypotenuse, and cosine relates adjacent to hypotenuse.

Real World Connection

In the Real World

Think about how your mobile phone's GPS app (like Google Maps or MapMyIndia) tells you exactly where you are and how to get to a friend's house in another city. It constantly uses trigonometry to calculate distances and angles between your current location, satellite signals, and your destination. Even ISRO uses trigonometry to track its satellites and plan their orbits!

Key Vocabulary

Key Terms

BEARING: The direction or an angle indicating direction, usually measured clockwise from North. | ANGLE OF ELEVATION: The angle from the horizontal upwards to an object. | ANGLE OF DEPRESSION: The angle from the horizontal downwards to an object. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | SOH CAH TOA: A mnemonic to remember trigonometric ratios (Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent).

What's Next

What to Learn Next

Next, you can explore the Law of Sines and Law of Cosines. These build on basic trigonometry to help solve problems involving non-right-angled triangles, which are very common in real-world navigation and engineering challenges. Keep learning, you're doing great!