Problem 59 - Algebra 2 and Trigonometry

Question

A ramp that is 12 feet long is used to reach a doorway that is 3.5 feet above the level ground. Find, to the nearest degree, the measure the ramp makes with the ground.

Step-by-Step Solution

Answer

The ramp makes an angle of approximately 17 degrees with the ground.

Step 1: Identify the right triangle components

The ramp forms the hypotenuse of a right triangle. The height from the ground to the doorway is the opposite side of the angle we need to find, and the length of the ramp is the hypotenuse. We are looking for the angle between the ground (adjacent side) and the ramp (hypotenuse).

Step 2: Use the sine ratio

To find the angle, we use the sine trigonometric function, which is defined as the ratio of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]Here, the opposite side is 3.5 feet and the hypotenuse is 12 feet.

Step 3: Calculate the sine of the angle

Substitute the given values into the sine function to find the angle:\[ \sin(\theta) = \frac{3.5}{12} = 0.2917\]

Step 4: Find the angle using inverse sine

To find the angle \(\theta\), take the inverse sine (\(\sin^{-1}\)) of 0.2917:\[ \theta = \sin^{-1}(0.2917)\]Using a calculator, \(\theta \approx 17\) degrees.

Key Concepts

Right Triangle

A right triangle is a fundamental shape in trigonometry, made unique by its characteristic 90-degree angle. This types of triangles are comprised of three sides: the hypotenuse and two other sides known as the legs.

• The hypotenuse is the longest side and opposite the right angle.
• One leg is adjacent to a non-right angle and runs along the base.
• The other leg is called the opposite side to a non-right angle.

In many practical applications, such as ramps or inclined planes, right triangles are used to understand spatial relationships. For example, in the context of ramps, the ramp itself often acts as the hypotenuse, providing a slope to ascend to a higher point. This setup allows us to apply trigonometric functions to solve for unknown angles or sides using known measures.

Sine Function

The sine function is one of the fundamental trigonometric functions used to relate the angles of a right triangle to the lengths of its sides.

It specifically deals with the opposite side and the hypotenuse of the triangle. In mathematical terms, the sine of an angle θ in a right triangle is defined as:

• Opposite side (specific to angle θ) over the hypotenuse.

Expressed in formulaic terms, this is written as:\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]With this function, if you know the lengths of these two sides, you can easily find the sine of the angle.
In ramp problems like the one discussed, we use this relation to determine the angle of inclination by substituting the known height of the ramp as the opposite side and the length of the ramp itself as the hypotenuse.

Inverse Sine

To find an angle when we know the sine value, we use the inverse sine function, often denoted as \( \sin^{-1} \) or arcsin. This function essentially reverses what the sine function does.

If you have calculated the sine of an angle, using the right sides of a triangle, the inverse sine helps you retrieve the angle measure. An important point to remember is that the result of an inverse sine calculation gives you the angle in degrees or radians, based on your calculator settings.

In the ramp scenario, after obtaining the sine value from the ratio \( \frac{3.5}{12} \), finding the inverse sine of 0.2917 allows us to determine the angle of the ramp with respect to the ground.
By understanding and applying the inverse sine function, we can solve equations backward, from known ratios to determine unknown angles, an essential method in solving right triangle problems.