Question

Write a program to determine the length of a ladder required to reach a given height when leaned against a house. The height and angle of the ladder are given as inputs. To compute length use: \\[ \text { length }=\frac{\text { height }}{\sin \text { angle }} \\] Note: The angle must be in radians. Prompt for an angle in degrees and use this formula to convert: \\[ \text {radians}=\frac{\pi}{180} \text { degrees } \\]

Short answer

Convert angle to radians, compute sine, then divide height by sine to find ladder length.

Step-by-step solution

Understand the Problem

We need to calculate the length of a ladder needed to reach a specified height when it is leaned against a house at a given angle. The given inputs are the height and the angle in degrees, and we want to find the ladder's length using the formula \( \text{length} = \frac{\text{height}}{\sin(\text{angle})} \). Note that the angle needs to be converted from degrees to radians.

Convert Angle from Degrees to Radians

The angle is initially given in degrees, but the length calculation requires it in radians. Use the formula: \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). This conversion adjusts the angle so it can be utilized in trigonometric functions like sine.

Calculate the Sine of the Angle

With the converted angle in radians, compute the sine of this angle. In most programming environments, this can be accomplished using a built-in function such as `math.sin()`.

Calculate the Length of the Ladder

Use the formula \( \text{length} = \frac{\text{height}}{\sin(\text{angle})} \) to determine the ladder's length. With the sine of the angle already computed, simply divide the given height by this sine value.

Write the Program

Create the program by scripting these steps. Use input statements to receive height and angle in degrees, apply the conversion to radians, calculate the sine, and then find the ladder length.

Key concepts

Trigonometry in Programming

Trigonometry plays a pivotal role in programming, especially in scenarios involving angles and distances. When calculating the length of a ladder that must reach a specific height, trigonometric functions become quite handy.
These functions allow us to determine lengths and angles within triangles. In programming, the sine function, denoted as \( \sin \), helps calculate the necessary ladder length. By using trigonometry, we can solve problems involving right-angle triangles by leveraging relationships between angles and sides.
This makes trigonometry invaluable in disciplines like game development, simulation, and modeling where spatial calculations are crucial.

Degree to Radian Conversion

When dealing with trigonometric functions in programming, angles often need to be converted into a different measure called radians. Radians are the standard unit of angular measure used in many mathematical functions because they simplify the calculations.

• Degrees are the common unit for angles in everyday usage.
• Radians provide a direct relationship between the arc length of a circle and its radius.

To convert degrees to radians, use the formula:
\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \] This conversion ensures that the trigonometric functions operate correctly, as the math libraries most programming languages use are based on radians.

Ladder Problem

The ladder problem is a classic example used to illustrate the application of trigonometry to real-world scenarios in programming. The aim is to determine the ladder's length needed to reach a certain height when leaned against a wall.
Imagine a right triangle formed by the house wall, the ground, and the ladder. The screw problem exploits the formula: \[ \text{length} = \frac{\text{height}}{\sin(\text{angle})} \] Here, the height is the opposite side of the angle, and the ladder is the hypotenuse. This requires computation using the angle’s sine value. Such exercises are invaluable for understanding how to translate mathematical formulas into actionable code.

Math Module in Python

Python's `math` module is a powerful tool that offers a wide range of mathematical functions. This module is essential when solving problems requiring complex calculations. Some key features include:

• `math.sin()`: computes the sine of a number in radians.
• `math.pi`: a constant representing the value of \( \pi \), crucial for converting degrees to radians.

Using the `math` module can significantly streamline your calculations in Python. Whether calculating angles, like in our ladder problem, or more advanced mathematical tasks, this module provides convenience and accuracy. It helps translate real-world problems into solvable programming challenges efficiently.