Problem 27 - Precalculus: Functions and Graphs

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\tan \theta=3.7$$

Step-by-Step Solution

Answer

(a) \(74.05^{\circ}\); (b) \(74^{\circ} 3^{\prime}\).

Step 1: Understanding the Problem

We need to find the acute angle \( \theta \) such that \( \tan \theta = 3.7 \). We're required to express \( \theta \) to the nearest \(0.01^{\circ}\) and to the nearest minute \(1^{\prime}\).

Step 2: Using the Inverse Tangent Function

To find \( \theta \), we apply the inverse tangent function: \( \theta = \tan^{-1}(3.7) \). This operation will give us the angle in degrees.

Step 3: Calculating \( \theta \) in Degrees

Using a calculator, compute \( \tan^{-1}(3.7) \). The result is approximately \(74.05^{\circ}\).

Step 4: Rounding to the Nearest Hundredth of a Degree

The value \(74.05^{\circ}\) is already rounded to two decimal places, or the nearest hundredth of a degree. No further actions are needed in this step.

Step 5: Converting Degrees to Degrees and Minutes

Since \(74.05^{\circ}\) is 74 degrees and a fraction of another degree, convert the fractional part \(0.05\) degrees into minutes. There are 60 minutes in a degree, so calculate: \(0.05 \times 60 = 3\), yielding \(74^{\circ} 3^{\prime}\).

Step 6: Rounding to the Nearest Minute

The angle is \(74^{\circ} 3^{\prime}\), which is already an integer minute. Therefore, the angle to the nearest minute is \(74^{\circ} 3^{\prime}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics. They allow us to find angles when we know the trigonometric ratio. For the tangent function, the inverse is denoted as \( \tan^{-1} \) or arctan.
For instance, if we have \( \tan \theta = x \), the angle \( \theta \) can be found using \( \theta = \tan^{-1}(x) \). This is crucial when you need to determine the angle that corresponds to a known tangent, like in our problem where \( \tan \theta = 3.7 \).
It's good to know that inverse trigonometric functions usually result in angles given in degrees or radians. In practical scenarios like this, a calculator will provide the angle in degrees unless stated otherwise. These calculations help us make sense of angles in many fields, from engineering to navigation.

Rounding Angles

Rounding angles is critical when we want precision while not diving into cumbersome detail. In this context, we often need to round angles to a specific decimal place or to the nearest minute.
Rounding to the nearest hundredth means looking at the third decimal place. If it's 5 or greater, we round up. For instance, an angle of \(74.051^{\circ}\) would be rounded to \(74.05^{\circ}\).
Rounding simplifies communication. Imagine pilots needing to convey flight angles; they need them precise yet simple. A clear decimal point or minute can reduce errors in such critical communications.

Converting Degrees to Minutes

Converting degrees to degrees and minutes is a practical skill, especially when dealing with precise measurements like navigation or astronomy. One degree has 60 minutes. This is similar to how a clock has 60 minutes per hour.
Start with the whole number part of your degrees, referring to degrees, and convert the fractional part. For \(74.05^{\circ}\), the ".05" part of the degree converts to minutes: \(0.05 \times 60 = 3\). This means \(74.05^{\circ} \) becomes \(74^{\circ} 3^{\prime}\).
Why does this matter? Well, in applications like mapping, every minute counts! By converting, we align clearer with how most navigational systems operate, providing exact positions.

Tangent Function

The tangent function, often abbreviated as \( \tan \), is one of the three primary trigonometric functions, alongside sine and cosine. The tangent of an angle in a right triangle represents the ratio of the opposite side to the adjacent side.
Symbolically, it's written as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). For example, if we have a right-angled triangle with a side of 3 units opposite our angle and 4 units adjacent, then \( \tan \theta = \frac{3}{4} \). These ratios are incredibly useful for calculations involving triangles.
In this problem, understanding \( \tan \theta = 3.7 \) means that for the specific triangle, the opposite side is 3.7 times the length of the adjacent side. This real-world interpretation allows us to apply this ratio to diverse situations like slope measurements or in physics to determine forces.