What is the Distance of an Object in Trigonometry?

S6-SA2-0109

Grade Level:

Class 10

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

Definition

What is it?

In trigonometry, the 'distance of an object' usually refers to calculating the length between two points or the measurement of a side of a right-angled triangle. We use trigonometric ratios (like sine, cosine, and tangent) along with known angles and sides to find these unknown distances.

Simple Example

Quick Example

Imagine you are flying a kite. You know the length of the string (hypotenuse) and the angle the string makes with the ground. To find how high the kite is from the ground (the vertical distance), you would use trigonometry.

Worked Example

Step-by-Step

PROBLEM: A ladder is leaning against a wall. The bottom of the ladder is 3 meters away from the wall, and it makes an angle of 60 degrees with the ground. How long is the ladder?

1. Identify the knowns: Distance from wall (adjacent side) = 3 m, Angle (theta) = 60 degrees. We need to find the length of the ladder (hypotenuse).
---2. Choose the correct trigonometric ratio: We have the adjacent side and want to find the hypotenuse. The cosine ratio relates adjacent and hypotenuse: cos(theta) = Adjacent / Hypotenuse.
---3. Set up the equation: cos(60 degrees) = 3 / Hypotenuse.
---4. Recall the value of cos(60 degrees): cos(60 degrees) = 0.5.
---5. Substitute the value: 0.5 = 3 / Hypotenuse.
---6. Solve for Hypotenuse: Hypotenuse = 3 / 0.5.
---7. Calculate the result: Hypotenuse = 6 meters.

ANSWER: The ladder is 6 meters long.

Why It Matters

Understanding distance in trigonometry is crucial for many fields. Engineers use it to design buildings and bridges, while scientists in space technology use it to calculate distances to planets or satellites. It's also vital for architects and surveyors in their daily work.

Common Mistakes

MISTAKE: Confusing which side is opposite, adjacent, or hypotenuse relative to the given angle. | CORRECTION: Always label the sides correctly based on the angle you are working with: the side opposite the right angle is the hypotenuse, the side next to the angle (not the hypotenuse) is adjacent, and the side across from the angle is opposite.

MISTAKE: Using the wrong trigonometric ratio for the given information (e.g., using sine when cosine is needed). | CORRECTION: Remember SOH CAH TOA. SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent). Match the ratio to the sides you know and the side you want to find.

MISTAKE: Incorrectly calculating the values of sin, cos, or tan for standard angles (0, 30, 45, 60, 90 degrees). | CORRECTION: Memorize the trigonometric values for these standard angles or know how to quickly derive them using a right-angled triangle sketch.

Practice Questions

Try It Yourself

QUESTION: A tower casts a shadow 15 meters long when the angle of elevation of the sun is 45 degrees. What is the height of the tower? | ANSWER: 15 meters

QUESTION: From a point on the ground 100 meters away from the base of a building, the angle of elevation to the top of the building is 30 degrees. Find the height of the building. (Use sqrt(3) = 1.732) | ANSWER: Approximately 57.74 meters

QUESTION: A boy is standing on a bridge 20 meters above a river. He sees a boat at an angle of depression of 30 degrees. How far is the boat from the point directly below the boy on the river surface? | ANSWER: Approximately 34.64 meters

MCQ

Quick Quiz

If a pole is 12 meters tall and casts a shadow of 12 meters, what is the angle of elevation of the sun?

30 degrees

45 degrees

60 degrees

90 degrees

The Correct Answer Is:

Since the height of the pole (opposite side) is equal to the length of the shadow (adjacent side), the tangent of the angle will be 1 (Opposite/Adjacent = 12/12 = 1). The angle whose tangent is 1 is 45 degrees.

Real World Connection

In the Real World

Trigonometry helps ISRO scientists track rockets and satellites by calculating their distances and trajectories using angles from ground stations. Similarly, surveyors use it to measure land boundaries for construction projects like new roads or metro lines, ensuring accurate distances and elevations.

Key Vocabulary

Key Terms

HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | OPPOSITE SIDE: The side directly across from a given acute angle in a right-angled triangle. | ADJACENT SIDE: The side next to a given acute angle in a right-angled triangle, which is not the hypotenuse. | ANGLE OF ELEVATION: The angle formed by the horizontal line of sight and the upward line of sight to an object. | ANGLE OF DEPRESSION: The angle formed by the horizontal line of sight and the downward line of sight to an object.

What's Next

What to Learn Next

Now that you understand how to find distances, you can explore 'Heights and Distances' problems, which apply these trigonometric concepts to real-life scenarios like measuring the height of a building or the distance between two objects. This will strengthen your problem-solving skills!