Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. Round answers to the nearest minute. $$1.74$$

Step-by-Step Solution

Answer

1.74 radians is approximately 99° 43'.

Step 1: Understand the Conversion Formula

Radians can be converted to degrees using the formula \[ ext{Degrees} = ext{Radians} \times \left(\frac{180}{ ext{π}}\right).\]We will apply this formula to the given radian measure of 1.74.

Step 2: Substitute Values into Formula

Insert 1.74 for radians into the formula: \[ ext{Degrees} = 1.74 \times \left(\frac{180}{ ext{π}}\right).\]Now calculate this expression.

Step 3: Perform Calculation to Find Degrees

Compute the conversion:\[ ext{Degrees} = 1.74 \times 57.2958 \approx 99.7141\]Thus, 1.74 radians is approximately 99.7141 degrees.

Step 4: Convert Decimal Degrees to Degrees and Minutes

To convert 99.7141 degrees to degrees and minutes, take the decimal part (0.7141) and multiply by 60 to convert to minutes:\[0.7141 \times 60 \approx 42.846\]Thus, we have approximately 99 degrees and 42.846 minutes.

Step 5: Round the Minutes to the Nearest Whole Number

Since 42.846 rounds up to 43, the final result is 99 degrees and 43 minutes.

Key Concepts

Angular Measurement

Angular measurement is a crucial concept within geometry and trigonometry, allowing us to quantify the size of an angle. Angles can be measured in different units, with degrees and radians being the most common:

• Degrees: Typically understood by most due to their usage in everyday scenarios, like navigation. There are 360 degrees in a full circle.
• Radians: This is the standard unit of angular measure used in many areas of mathematics. One radian is defined by the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Radians provide a natural connection between angle measurements and other mathematical concepts like trigonometric functions. This makes them especially important in higher-level mathematics and physics.

Understanding how to convert between these two units allows for a better grasp of both mathematical theories and practical applications. It forms the basis of many calculations, including those involving circular motion and wave functions.

Degree and Minute Conversion

When discussing degree and minute conversion, we are focusing on translating angular measurements into a more refined format. This breakdown allows for greater precision when dealing with angles.

Similar to how time is split into hours, minutes, and seconds, degrees are sometimes broken into degrees, minutes, and seconds, where:

• Degrees: The main unit, indicated by a ° symbol.
• Minutes: A further division of a degree, with one degree comprising 60 minutes (').
• Seconds: Each minute is further divided into 60 seconds (''). However, this level of detail is generally not necessary unless extreme precision is required.

In practice, once an angle is determined in decimal degrees from radians, the fractional part is multiplied by 60 to convert into minutes. As shown in the original problem, when 99.7141 degrees is broken down, the .7141 becomes approximately 43 minutes upon multiplication and rounding.

This method provides a convenient numerical format that can be easily communicated and used, especially in fields such as navigation and astronomy.

Radian Measure

Radian measure is fundamental in the world of mathematics, offering a direct link between the geometry of a circle and mathematical analysis. It often simplifies many mathematical equations and expressions.

A radian is defined in terms of pi (π), which itself is an essential constant in mathematics.

• One radian equals approximately 57.2958 degrees. This is derived from the fact that the circumference of the full circle (2π radians) equals 360 degrees.
• This relationship allows for converting any radian measurement to degrees using the formula: \[ \text{Degrees} = \text{Radians} \times \left(\frac{180}{\pi}\right). \]
• This is crucial for solving problems in trigonometry, calculus, and physics, where angles need to be expressed in the most versatile manner.

By understanding and using radians, students can gain a deeper appreciation of mathematical phenomena and real-world applications, where angles are not merely numbers but pivotal components of complex calculations.