Problem 73 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 190^{\circ} $$

Step-by-Step Solution

Answer

Therefore, \(190^{\circ}\) is approximately \(3.33\) radians or \(\frac{19\pi}{18}\) radians.

Step 1: Understand the relationship between degrees and radians

Use the relationship \(180^{\circ} = \pi\) radians to convert degrees to radians.

Step 2: Convert the given degrees into radians using the relationship from step 1

In order to convert the degree measurement into radians, you need to substitute the degree measurement into the relationship. Therefore, \(190^{\circ} = 190 \times \frac{pi}{180} radians\)

Step 3: Simplify the expression

Simplify the result by multiplying 190 and \(\frac{pi}{180}\), to obtain an expression in terms of \(\pi\). Therefore, the result in terms of \(\pi\) is approximately \(\frac{19\pi}{18}\) radians.

Step 4: Convert the radian measure into a rounded decimal

To convert the radian measurement to a rounded decimal, multiply \(\frac{19\pi}{18}\) by the decimal approximation of \(\pi\) (~3.14), and round to the nearest hundredth, obtaining approximately 3.33 radians.

Key Concepts

Degrees to Radians

When it comes to measuring angles, two units are widely used: degrees and radians. Degrees are often used in everyday contexts, like reading a protractor, while radians are used primarily in mathematics and physics. Converting degrees to radians is straightforward once you understand the fundamental relationship between the two. This relationship is:

• One full circle is 360 degrees.
• The same circle in radians is expressed as \( 2\pi \) radians.

To convert degrees to radians, use this formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]. This formula comes from dividing the number of degrees in a full circle by the number of radians in a full circle: 360 divided by \( 2\pi \). Substitute the degree number you wish to convert into this formula, and you'll obtain the angle in radians.

Mathematical Conversion

The process of mathematical conversion between different units is fundamental in making calculations that are more appropriate to the problem you're solving. Conversion can often simplify the arithmetic or reveal more natural units for a given context. In mathematical conversion from degrees to radians, as seen in the example where 190 degrees is converted to radians, the following steps are key:

• Start by using the conversion factor \( \frac{\pi}{180} \).
• Multiply this factor with the angle in degrees.
• Simplify the expression to get the result in terms of \( \pi \).

For example, converting \( 190^{\circ} \) into radians involves multiplying: \[ 190 \times \frac{\pi}{180} = \frac{19\pi}{18} \]. This gives the angle in a more universal mathematical language, radians, which are easier to use in calculus and trigonometry.

Pi in Mathematics

The constant \( \pi \) (pi) is one of the most important numbers in mathematics, representing the ratio of a circle's circumference to its diameter. Pi is crucial in geometry, trigonometry, calculus, and many areas of applied mathematics. It is an irrational number, which means it cannot be expressed exactly as a simple fraction, and it's approximate value is 3.14159, but typically rounded to 3.14 for many calculations.

• Pi is not only pivotal in area and circumference calculations but also in the conversion between degrees and radians.
• In the degree-radian conversion formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \), pi bridges the gap between linear and angular measurements.
• When expressing radians, keeping pi in the expression, such as \( \frac{19\pi}{18} \), retains precision.

Using pi maintains precision in theoretical calculations, whereas its decimal approximation is suitable for practical calculations requiring numerical values.