Problem 15 - Algebra 2 and Trigonometry

Question

A ladder that is 15 feet long is placed so that it reaches from level ground to the top of a vertical wall that is 13 feet high. a. Use the Law of Sines to find the angle that the ladder makes with the ground to the nearest hundredth. b. Is more than one position of the ladder possible? Explain your answer.

Step-by-Step Solution

Answer

a) The angle is approximately 60.92 degrees. b) No, only one ladder position is possible.

Step 1: Understanding the Given Information

We have a ladder of length 15 feet leaning against a wall that's 13 feet high. We must find the angle between the ladder and the ground using trigonometry principles.

Step 2: Applying the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Here, the side opposite to angle \( C \) is 13 feet (wall height) and side \( a \) (ladder) is 15 feet. Assuming \( C \) is the angle between the ground and the ladder (angle to find), we start with \( \sin(A) = \frac{13}{15} \). We also know angle \( A \) at the wall is 90 degrees, so this is not needed for calculation.

Step 3: Calculating the Angle C Using Inverse Sine

To find angle \( C \), use: \( \sin C = \frac{13}{15} \). Hence, \( C = \arcsin\left(\frac{13}{15}\right) \approx 60.92^\circ \). Calculators are used to compute this to the nearest hundredth.

Step 4: Determining Other Possible Ladder Positions

Given this is a problem involving a right triangle with side lengths defined by the ladder and the wall, another position is not possible as the ladder must remain a hypotenuse. Consequently, only this solution exists.

Key Concepts

Law of Sines

The Law of Sines is a fundamental concept in trigonometry used to solve various types of triangle problems, including those involving non-right triangles. It states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant across all three sides:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

In the exercise, the Law of Sines helps us find the angle between the ladder and the ground. We know one of the angles is 90 degrees, simplifying calculations because this angle does not affect the application of the law directly. Instead, we use it to relate the known side lengths with the unknown angle using sine values. Though typically applicable for general triangles, here it assists by linking the known dimensions of the triangle formed by the ladder and wall.

Right Triangle

Right triangles are a staple in geometry and trigonometry. They have one angle that is exactly 90 degrees and two other angles that are acute. In this exercise, the ladder, wall, and ground form a right triangle when placed correctly. The ladder represents the hypotenuse since it is the longest side, opposite the right angle.

• The vertical wall is one leg of the triangle, measuring 13 feet.
• The ground on which the ladder rests is the other leg.

Right triangles make it straightforward to apply trigonometric principles due to their inherent ratios like sine, cosine, and tangent. In calculations, positions of angles and sides are quite predictable, simplifying tasks such as determining heights, distances, or angles, as in this scenario. Additionally, its uniqueness due to the right angle limits solutions to most setups to a single configuration, as emphasized in the solution.

Inverse Sine

Inverse sine, often denoted as \( \arcsin \), is the process used to find an angle when given the sine of that angle. It is the reverse of the sine function and allows us to compute an angle when we know the ratio of the opposite side to the hypotenuse.

• For example, if \( \sin \theta = x \), then \( \theta = \arcsin(x) \).

In the exercise, the inverse sine is crucial when calculating the angle between the ladder and the ground. With the known ratio of the wall height (13 feet) to the ladder's length (15 feet), \( \sin C \) is found using the equation \( \sin C = \frac{13}{15} \). By applying the inverse sine function, we determine that angle \( C \) approximately equals 60.92 degrees. Calculators or mathematical software are typically used to find this angle with precision, ensuring students interpret results clearly and accurately without manual approximation errors.