Problem 62 - Algebra and Trigonometry

Question

A kite flies at a height of 35 feet when 60 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 string makes with the ground when the kite is 35 feet high and 60 feet of string is out is approximately 30.1 degrees.

Step 1: Identify Knowns

The length of the string (the hypotenuse) is 60 feet, and the height of the kite (the opposite side) is 35 feet.

Step 2: Apply the Tangent Formula

Use the formula for the tangent of an angle in a right triangle. The angle in question is between the ground and the kite string, so we can call it θ. The formula is tan(θ) = opposite/hypotenuse. Plugging the known values into the formula gives tan(θ) = 35 / 60 = 0.5833.

Step 3: Calculate the Angle

Since we have the tangent of the angle, we will use the inverse tangent or arctan function to find the angle in radians, and then convert it to degrees. So, θ = arctan(0.5833).

Step 4: Convert Rad to Degrees

Convert the measure of θ from radians to degrees by multiplying by (180/π).

Step 5: Round the Result

Round the resulting angle to the nearest tenth of a degree as instructed in the problem statement.

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle where one angle is exactly 90 degrees. It is composed of three sides: the hypotenuse, opposite side, and adjacent side. The hypotenuse is always the longest side and is opposite the right angle. In many practical scenarios like this one, where you are determining the position of a kite, understanding the layout of a right triangle is crucial. Knowing the lengths of any two sides lets you easily calculate the unknowns if you know the relationships like the trigonometric ratios. Right triangles are foundational in trigonometry, helping to solve real-life problems like measuring distances and angles.

Exploring the Tangent Function

The tangent function is one of the three primary trigonometric functions, and it's usually denoted as tan(θ). In the context of a right triangle, the tangent of an angle θ is defined as the ratio of the length of the opposite side to the length of the adjacent side: \( an(θ) = \frac{\text{opposite}}{\text{adjacent}} \). However, when working with the hypotenuse and opposite as given, we use a rearranged understanding as follows: \( an(θ) = \frac{\text{opposite}}{\text{hypotenuse}} \).

• It helps measure how `steep` an angle is.
• The tangent function is useful when you know two sides of a right triangle and need to find an angle.

The tangent function is essential for angle determination in scenarios of navigation and astronomy.

Using Inverse Tangent for Angle Finding

After figuring out the ratio using the tangent function, we often need to find the angle that corresponds to this ratio. This is where the inverse tangent function, or arctan, comes into play.

• The inverse tangent, denoted as \( \tan^{-1} \) or \( \arctan \), reverses the tangent process, taking a ratio as input and giving the measure of the angle as output.
• It is especially useful when you have the tangent value and need to convert it back to an angle measurement.

In practical applications, especially in this problem of determining the kite's angle to the ground, using the inverse tangent allows us to find the precise degree of angle needed to solve the kite's position.

Angle Calculation Techniques

Calculating angles using trigonometric functions involves a few straightforward steps. Once you have the tangent ratio, you use the inverse tangent function to find the angle in radians. Here's a step-by-step observation of the process:

• First, calculate the tangent ratio: for our scenario, \( an(θ) = \frac{35}{60} = 0.5833 \).
• Then, apply the inverse tangent function: \( θ = \arctan(0.5833) \).
• The output will give you the angle in radians, which you will need to convert to degrees because degrees are more commonly used in navigation and everyday measurements.
• Multiply by \( \frac{180}{π} \) to convert radians to degrees.
• Finally, round the result to the nearest tenth as is typical in angle calculations, making it practical and easy to understand.

This problem-solving technique is applicable in various fields like engineering, physics, and everyday tasks such as architecture or positioning objects.