Problem 45 - Precalculus : Building Concepts and Connections

Question

The base of a railing for a staircase makes an angle of \(x\) degrees with the horizontal. Let \(d(x)\) be the horizontal distance between the two ends of the base of the railing. If point \(L\) on the railing is 5 feet higher than point \(M,\) find the positive number \(A\) such that \(d(x)=A\) cot \(x .\) Then use your function to find the length of the base of the railing if \(x=35^{\circ}\).

Step-by-Step Solution

Answer

The length of the base of the railing when \(x = 35^{\circ}\) is \(d(35) = A \cdot cot (35^{\circ})\), with \(A\) as the value found in Step 2.

Step 1: Setting Up the Equation

Given, point L on the railing is 5 feet higher than point M, which forms a right triangle with the base of the railing \(d(x)\) and the railing (hypotenuse). From trigonometry, we know that the cotangent of an angle is the ratio of the base to height in a right triangle. Hence, if we denote the length of railing as \(A\), then it can be expressed as \(A=d(x) \cdot sin(x)\).

Step 2: Computing the Value of A

Using trigonometric relations, convert \(A = d(x) \cdot sin(x)\) to \(A = d(x)\cdot cot(x)\). Hence, \(A\) is equal to the length of the railing when \(x = 90^{\circ}\) since cotangent of \(90^{\circ}\) equals 0. The final established function becomes \(d(x) = A \cdot cot(x)\).

Step 3: Finding the Length of the Base of the Railing

After determining \(A\), use the function \(d(x) = A \cdot cot(x)\) to find the length of the base of the railing when the \(x = 35^{\circ}\). Plug \(35^{\circ}\) in place of \(x\) and compute the value of \(d(x)\). A is the value you found in step 2.

Key Concepts

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry and are critical in understanding the relationships between angles and sides of triangles. To put it briefly, they define how the dimensions of a triangle change with its angles. The key functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Each function takes an angle as input and returns a specific ratio of two sides of a right triangle. For instance, the sine function represents the ratio of the length of the side opposite the angle to the hypotenuse, while the cosine function represents the ratio of the adjacent side to the hypotenuse. Understanding these functions is vital in various fields, including physics, engineering, and even computer science.

Right Triangle Trigonometry

Right triangle trigonometry is specifically concerned with the properties and ratios of the sides of triangles that have one 90-degree angle, known as right triangles. The three main trigonometric ratios—sine, cosine, and tangent—are based on the angles and sides of a right triangle. These ratios allow us to determine the lengths of sides when we know the measurement of at least one angle and one side, which proves useful in numerous practical situations.

In our railing example, the triangle formed has a height (opposite the angle x), a base (adjacent to angle x), and a hypotenuse (the railing itself). By applying the trigonometric ratios, one can find out the necessary lengths and distances to solve real-world problems.

Trigonometric Ratios

Trigonometric ratios are specific numbers that provide a quantifiable relationship between the angles of a triangle and the lengths of its sides. To elaborate:

• The sine of an angle is the ratio of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the opposite side to the adjacent side.

Less commonly used but equally important are the reciprocals of these functions: the cosecant, secant, and cotangent, which are the ratios of the hypotenuse to the opposite side, hypotenuse to the adjacent side, and adjacent side to the opposite side, respectively.

The cotangent function played a central role in solving the staircase railing problem. By understanding that the cotangent of an angle in a right triangle is the length of the adjacent side divided by the opposite side, we were able to derive the function for the horizontal distance of the railing base as a function of the angle x, proving how indispensable these trigonometric ratios are for solving geometry problems.