Problem 57 - Calculus - AP Edition

Question

The hypotenuse of a right triangle is known to be 10 in exactly, and one of the acute angles is measured to be \(30^{\circ}\) with a possible error of \(\pm 1^{\circ}\). (a) Use differentials to estimate the errors in the sides opposite and adjacent to the measured angle. (b) Estimate the percentage errors in the sides.

Step-by-Step Solution

Answer

The errors in the sides are approximately \( \pm 0.15 \) and \( \mp 0.0873 \). Percentage errors are \(3\%\) and \(1.01\%\).

Step 1: Understand the Context of the Problem

We have a right triangle with a fixed hypotenuse of length 10 inches, and one of the acute angles is approximately \(30^{\circ}\), with a possible deviation of \(\pm 1^{\circ}\). We need to find the errors in the opposite and adjacent side lengths caused by this angle error.

Step 2: Express Side Lengths Using Trigonometric Functions

Given a right triangle with hypotenuse \( h = 10 \) and angle \( \theta = 30^{\circ} \), the opposite side can be computed using \( \, \sin(\theta) = \frac{opposite}{hypotenuse} \, \). Thus, \( \, opposite = h \cdot \sin(\theta) = 10 \cdot \sin(30^{\circ}) = 10 \cdot 0.5 = 5 \) inches.The adjacent side is found using the cosine function: \( \, adjacent = h \cdot \cos(\theta) = 10 \cdot \cos(30^{\circ}) = 10 \cdot \sqrt{3}/2 \approx 8.66 \, \text{inches}.\)

Step 3: Apply Differentials to Estimate Errors

Differentials provide us a way to estimate the change in a function due to a small change in the input.For the opposite side: Let \( f(\theta) = 10 \cdot \sin(\theta) \). The differential \( df = f'(\theta) \, d\theta = 10 \cdot \cos(30^{\circ}) \, d\theta \). For \( d\theta = \pm \frac{\pi}{180} \) (converting \( \pm 1^{\circ} \) to radians), the error in the opposite side is: \[ df = 10 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\pi}{180} \approx \pm 0.15. \]For the adjacent side: Let \( g(\theta) = 10 \cdot \cos(\theta) \). The differential \( dg = g'(\theta) \, d\theta = -10 \cdot \sin(30^{\circ}) \, d\theta \), leading to an error for \( d\theta = \pm \frac{\pi}{180} \):\[ dg = -10 \cdot 0.5 \cdot \frac{\pi}{180} \approx \mp 0.0873. \]

Step 4: Calculate the Percentage Errors

Percentage error for a quantity is given by \( \frac{\text{absolute error}}{\text{original value}} \times 100\% \).For the opposite side: \( \frac{0.15}{5} \times 100\% = 3\% \).For the adjacent side: \( \frac{0.0873}{8.66} \times 100\% \approx 1.01\% \).

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential for solving problems involving right triangles. In a right triangle, these functions relate the angles to the lengths of the sides. The main trigonometric functions used frequently are sine, cosine, and tangent.

• Sine (sin) is defined as the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) equals the ratio of the opposite side to the adjacent side.

In our exercise, the angle of interest is given as 30 degrees. By applying the sine function, we calculate the opposite side, and using the cosine function, we compute the adjacent side. Knowing these basics help us to effectively solve for the unknown sides when the angle and one side (the hypotenuse) are known.

Error Estimation

Error estimation is crucial when dealing with measurements, as they often have limits of precision. In calculus, differentials are used to estimate the changes in function outputs due to slight changes in input values. This is particularly useful when assessing errors in calculated values due to measurement inaccuracies. In the problem, we use differentials to estimate how a tiny error in the angle measurement affects the length of the sides of a triangle. We convert a potential error of ±1 degree into radians, as calculus formulas require. The formulas help us compute the differential, giving us an estimate of how much each side's length could be off.
This approach aids in making informed decisions based on the reliability of measured or calculated values, by evaluating how a small error in input can propagate through to the result.

Right Triangle Geometry

The right triangle is a fundamental concept in geometry and involves a 90-degree angle. With a fixed hypotenuse and angles, right triangles allow us to utilize trigonometric functions to solve various geometric problems. In this exercise, understanding right triangle geometry is key to determining the opposite and adjacent side lengths concerning the hypotenuse of 10 inches. Calculating the exact side lengths involves applying the known angles and trigonometric identities which leverage constants and properties unique to 30-60-90 triangles.

• The hypotenuse is the longest side, opposite the right angle.
• The angle of 30 degrees provides a specific ratio between the sides thanks to known properties.
• Understanding these ratios simplifies the calculation and leads to easier error checking.

Focusing on such geometric principles allows us to accurately solve for unknowns in triangles, any changes in angles or sides are subsequently scrutinized for precision using error analysis.