Problem 27 - Algebra and Trigonometry with Analytic Geometry

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) \(74.74^{\circ}\), (b) \(74^{\circ} 45^{\prime}\)."

Step 1: Understanding the Problem

We need to find an angle \( \theta \) such that \( \tan \theta = 3.7 \). We'll approximate \( \theta \) first to the nearest \( 0.01^{\circ} \) and then to the nearest minute (\( 1^{\prime} \)).

Step 2: Calculating \( \theta \) Using Arctangent

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

Step 3: Approximating \( \theta \) to Nearest Degree (Part a)

Use a calculator to find \( \theta = \tan^{-1}(3.7) \). This gives approximately \( 74.744^{\circ} \). Approximating to the nearest \( 0.01^{\circ} \), we get \( 74.74^{\circ} \).

Step 4: Approximating \( \theta \) to Nearest Minute (Part b)

Convert the decimal degrees from Step 3 to degrees and minutes. \( 0.744 \times 60 = 44.64 \). Therefore, \( 74^{\circ} 44.64^{\prime} \) approximates to the nearest minute as \( 74^{\circ} 45^{\prime} \).

Key Concepts

Inverse Tangent Function

The inverse tangent function is a vital part of trigonometry. It helps us find an angle when given the value of its tangent. This function is denoted by either \( \tan^{-1} \theta \) or \( \arctan \theta \). Imagine you're looking to find an angle whose tangent value is known. By using the inverse tangent, you can convert that tangent value back into an angle. This process is useful in various mathematical problems where angles and slopes intersect. For instance, if you're told that \( \tan \theta = 3.7 \), you're actually calculating the angle \( \theta \) corresponding to this tangent value using the inverse tangent function. The key benefit here is that the inverse tangent directly provides the angle in degrees. This makes it much easier to work with and visualize in most trigonometric scenarios. By understanding how to use this tool, you'll find it much more straightforward to handle problems involving angles and slopes.

Acute Angle Approximation

When working with angles, especially in trigonometry, approximating them can often lead to simpler calculations. Angles can be quite exact, sometimes needing to be very precise, down to fractions of a degree. In this exercise, we approximated an acute angle \( \theta \) to the nearest \( 0.01^{\circ} \).Why is this important? It's because real-world calculations don't always need the maximum level of precision that mathematics can offer. Here's how acute angle approximation works:

• First, find the angle with a scientific calculator by using the inverse tangent function.
• Convert the precise value into a more manageable form, like rounding to two decimal places.

For example, the calculated angle \( 74.744^{\circ} \) becomes \( 74.74^{\circ} \) when rounded to the nearest hundredth. This type of approximation keeps your results both accurate and practical, saving time and effort while solving problems.

Degree-Minute Conversion

Degrees and minutes are part of the way we measure angles in trigonometry. Converting an angle from degrees to degrees and minutes is a useful skill, especially when you want to express it in a more traditional format.Here's how you can convert a decimal degree to degrees and minutes:

• Start with the angle in decimal degrees, such as \( 74.744^{\circ} \).
• Take the fractional part (after the decimal), which is \( 0.744 \), and multiply it by 60, since there are 60 minutes in a degree.
• The product \( 0.744 \times 60 = 44.64 \) tells you the minutes.
• Round the minutes to the nearest whole number if necessary. Here, \( 44.64 \) gets rounded to \( 45 \).

Thus, \( 74.744^{\circ} \) is converted to \( 74^{\circ} 45^{\prime} \). This representation is often easier when measuring angles in various fields such as navigation and geography.