Calculating the Angle of Inclination for a Kite String

The problem describes a kite flying at a certain height with a given length of string. We need to find the angle that the kite string makes with the horizontal ground. This situation can be modeled using a right-angled triangle.

Understanding the Setup

Imagine the following:

• The kite is at a point above the ground.
• The height of the kite is the vertical distance from the kite to the ground.
• The kite string is the line segment connecting the kite to the point on the ground where the string is held.
• The horizontal ground forms the base of the triangle.

These three lines form a right-angled triangle where:

• The height of the kite is the side opposite to the angle of inclination.
• The length of the string is the hypotenuse (the longest side, opposite the right angle).
• The horizontal distance from the person holding the string to the point directly below the kite is the adjacent side.

We are given:

• Height of the kite (Opposite side) = 50 m
• Length of the string (Hypotenuse) = 100 m

We need to find the angle of inclination, let's call it $\theta$, which is the angle between the string and the horizontal ground.

Using Trigonometry to Find the Angle

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

Mathematically, this is written as:

\(\text{sin}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)

Calculation Steps

Substitute the given values into the sine formula:

\(\text{sin}(\theta) = \frac{50 \, \text{m}}{100 \, \text{m}}\)

\(\text{sin}(\theta) = \frac{1}{2}\)

To find the angle \(\theta\), we need to find the inverse sine (arcsin or $\text{sin}^{-1}$) of \(\frac{1}{2}\).

\(\theta = \text{sin}^{-1}\left(\frac{1}{2}\right)\)

We need to recall the standard trigonometric values for common angles. The angle whose sine is \(\frac{1}{2}\) is 30 degrees.

\(\theta = 30^\circ\)

Thus, the inclination of the kite string to the horizontal ground is 30 degrees.

Checking the Options

Let's compare our calculated angle with the given options:

• A. 90°
• B. 45°
• C. 60°
• D. 30°

Our calculated angle is 30°, which matches option D.

Summary of Calculation

The problem involves a right triangle formed by the kite's height, the string, and the ground. Using the given height (opposite side) and string length (hypotenuse), we used the sine function to find the angle of inclination. \(\text{sin}(\theta) = \frac{50}{100} = \frac{1}{2}\). The angle whose sine is $\frac{1}{2}$ is 30 degrees.

Revision Table: Common Angles & Sine Values

Additional Information: Angle of Elevation

The angle of inclination of the kite string to the horizontal ground is also known as the angle of elevation from the point on the ground to the kite. The angle of elevation is the angle formed by the horizontal line and the line of sight to an object above the horizontal line. In this case, the horizontal line is the ground, and the line of sight is the kite string.

Understanding angles of elevation and depression is crucial in solving problems involving heights and distances using trigonometry.

This problem demonstrates a practical application of basic trigonometry, specifically the sine ratio, in calculating angles in real-world scenarios.