Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. Round answers to the nearest minute. $$-4$$

Step-by-Step Solution

Answer

-229° 11'

Step 1: Understanding the Conversion

To convert from radians to degrees, recall that a full circle in radians is equal to \(2\pi\) and in degrees, it is equal to \(360^\circ\). Therefore, the conversion formula from radians to degrees is \( \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \).

Step 2: Applying the Formula

Given the radian measure is \(-4\), apply the conversion formula: \[ \theta_{\text{degrees}} = -4 \times \frac{180}{\pi} \].

Step 3: Calculating Degrees

Calculate the degrees by performing the multiplication:\[ \theta_{\text{degrees}} = -4 \times \frac{180}{3.14159265} \approx -229.183 \].

Step 4: Converting to Degrees and Minutes

Separate the integer part from the decimal part. Here, the integer part is \(-229\) degrees. The decimal part \(0.183\) must be converted into minutes: \[ 0.183 \times 60 \approx 10.98 \].

Step 5: Rounding to the Nearest Minute

Round \(10.98\) to the nearest whole number, yielding \(11\). Thus, the final answer is \(-229^\circ\, 11'\).

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the lengths and angles of triangles. It plays a crucial role in many fields, such as physics, engineering, and architecture. The three primary functions in trigonometry are sine, cosine, and tangent. These functions help in determining the ratios of a right triangle's sides, given certain angles. Understanding trigonometry is essential for solving problems involving angles and distances. Moreover, trigonometry extends beyond right triangles and considers angular measures in the broader context of a circle, utilizing radian measures as a systematic approach. Radian is a fundamental angle measure in trigonometry, directly related to the circle's radius. It provides a natural way of describing angles, which simplifies many calculations in mathematics and physics.

Angle Measurement

Angle measurement is all about understanding how much space an angle takes up between two intersecting lines. There are two primary units of measurement for angles: degrees and radians.A degree is a way to express angles where a full circle is divided into 360 equal parts. Radians, on the other hand, measure the angle based on the circle's radius. In the radian system, a full circle is equivalent to \(2\pi\) radians. This makes radians particularly useful in calculus and trigonometry.In converting from radians to degrees, you use the formula:

• \( \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \)

This formula helps transform the radian measure of angle into degrees, making it easier to interpret and use in various applications.

Degree and Minute Conversion

The degree and minute conversion process involves turning a decimal degree measurement into degrees and minutes. This form of representation is especially useful in geographic and astronomical settings, where precise angle measurement is necessary.First, when you have an angle measurement like \(-229.183\) degrees from a radian conversion, begin by identifying the integer part, which is \(-229\) degrees. Then, focus on the decimal portion \(0.183\). To convert decimals into minutes:

• Multiply the decimal by 60 since there are 60 minutes in a degree.
• \(0.183 \times 60 \approx 10.98\)

Finally, round this to the nearest whole number, which gives \(11\) minutes. Therefore, the angle in degrees and minutes becomes \(-229^\circ 11'\). This conversion proves essential when precise angle measures are needed.