Problem 48 - Thinking Mathematically

Question

A kite flies at a height of 30 feet when 65 feet of string is out. If the string is in a straight line, find the angle that it makes with the ground. Round to the nearest tenth of a degree.

Step-by-Step Solution

Answer

The angle that the kite string makes with the ground is approximately 24.9 degrees when rounded to the nearest tenth.

Step 1: Identify the given data and the unknown

We are given that the kite is 30 feet high from the ground and the length of the string is 65 feet. The unknown is the angle that the string makes with the ground.

Step 2: Apply the tangent ratio

The tangent of the angle (\(\theta\)) we are trying to find can be calculated by the ratio height/string length. So, \(\tan(\theta) = \frac{30}{65}\)

Step 3: Find the angle using an inverse tangent function

Using inverse tangent function (arctan or tan^(-1)), we can find the angle. Therefore, \(\theta = \tan^{-1}(\frac{30}{65})\) The arctan can be found by using a scientific calculator.

Step 4: Convert the angle to degrees and round off to the nearest tenth

After finding the angle in radians using a calculator, the value should be converted to degrees as the question is asking for the answer in degrees. If the calculator is in degree mode, this step can be ignored. Lastly, round off the answer to the nearest tenth.

Key Concepts

Tangent Ratio

The tangent ratio is a fundamental concept in trigonometry that is used to relate the angles in a right triangle to the lengths of its sides. Specifically, the tangent of an angle in a right triangle is the ratio of the length of the side opposite to the angle (the opposite side) to the length of the side adjacent to the angle (the adjacent side).

This can be expressed with the formula:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In practical terms, if you were to look at a right triangle with one of the non-right angles labelled \( \theta \), and side lengths corresponding to 'opposite' and 'adjacent' regarding that angle, the tangent of \( \theta \) would give you a specific value. This value can be used to solve various geometric problems, including determining missing side lengths or angles.

Inverse Tangent Function

The inverse tangent function, often denoted as \( \tan^{-1} \) or arctan, serves as the reverse operation of the tangent function. While the tangent function gives you a ratio from an angle, the inverse tangent function gives you an angle from a ratio.

When you have a ratio of the opposite side to the adjacent side and need to find the particular angle that produces this ratio, the inverse tangent function is your tool of choice. For example:
\[ \theta = \tan^{-1}(\text{ratio}) \]
The result will give you the measure of the angle \( \theta \) in radians by default. To express this angle in degrees, which is a more common unit of measurement in geometry, a calculator with the capability to convert radians to degrees, or applying the conversion factor \( \frac{180}{\pi} \), is necessary.

Angle Calculation

Angle calculation is a pivotal aspect of trigonometry and is essential for solving many types of geometric problems. To calculate an angle in a right triangle when given two side lengths, one can use trigonometric ratios like the tangent ratio. After setting up the appropriate ratio—such as opposite over adjacent—the inverse trigonometric function can be applied to find the angle's measure.

In the context of the exercise involving the kite, once we have the tangent ratio using the height and string length as 'opposite' and 'adjacent' respectively, we use the inverse tangent function to find the angle measurement. Calculators with trigonometric functions are widely used to compute these values precisely and quickly. Remember that angles can be reported in either radians or degrees, and it's crucial to know which unit the problem requires and convert if necessary.

Mathematical Problem-Solving

Mathematical problem-solving involves a systematic approach to tackling mathematical questions. In the context of trigonometry and angle measurements, this typically involves several key steps. First, clearly identify the given information and what needs to be found. Second, determine the appropriate mathematical principles or formulas to use—in this case, the tangent ratio and the inverse tangent function.

Once the appropriate formula is selected, it is applied, often using a calculator to handle the trigonometric calculations. If the initial calculation does not directly provide the answer in the required unit, further steps may be necessary, such as converting radians to degrees. Finally, ensure to round off the answer according to the given instructions to arrive at the correct solution.