Problem 2 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 150^{\circ} $$

Step-by-Step Solution

Answer

The conversion of 150 degrees to radians is \(\frac{5\pi}{6}\) or approximately 2.62 (when expressed as a decimal).

Step 1: Conversion Relationship

Recognize the conversion relationship between degrees and radians. We know that \(1^{\circ} = \frac{\pi}{180}\) radians.

Step 2: Apply Conversion

Next, utilize the conversion relationship to convert 150 degrees to radians. To do this, multiply 150 by the fraction \(\frac{\pi}{180}\). Which gives us \(\frac{150\pi}{180}\)

Step 3: Simplify Result

Simplify the result by canceling out common factors in the numerator and the denominator. When we reduce the fraction \(\frac{150\pi}{180}\), we get \(\frac{5\pi}{6}\)

Step 4: Calculate Decimal

If the exercise also requires a decimal representation, we need to calculate the value of \(\frac{5\pi}{6}\) by employing the known approximate value of \(\pi\) as 3.14. This results in approximately 2.62.

Key Concepts

Degree to Radian Conversion

When converting angles from degrees to radians, we're essentially changing units. This is like changing inches to centimeters. The key is in understanding the conversion factor.

• One complete circle is 360 degrees but can also be expressed as \(2\pi\) radians.
• Thus, \(1^{\circ} = \frac{\pi}{180}\) radians.

To convert 150 degrees to radians, multiply 150 by \(\frac{\pi}{180}\). This step converts the angle into the radian unit, which is more convenient for many mathematical calculations.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, which makes them easier to work with. It's like reducing 8/12 to 2/3.
In our conversion, we arrived at \(\frac{150\pi}{180}\). To simplify:

• Find the greatest common divisor (GCD) of 150 and 180, which is 30.
• Divide both the numerator and the denominator by 30.
• This gives us the simplified fraction \(\frac{5\pi}{6}\).

Simplifying makes it clearer and sometimes easier to see relationships between numbers.

Mathematical Calculations

Performing mathematical calculations with fractions and decimals is an essential skill in math. Once you have simplified your fraction, you might need to convert it to a decimal.

• Using \(\pi \approx 3.14\), calculate \(\frac{5\pi}{6}\).
• This involves simple multiplication and division: \(5 \times 3.14 = 15.7\), and then dividing by 6 gives approximately 2.62.

These steps help verify that the conversion is not just symbolically correct but also numerically accurate.

Use of Pi in Mathematics

Pi (\(\pi\)) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it has an infinite number of non-repeating decimals.

• In calculations, we often use \(\pi\) symbolically to maintain precision.
• In practical applications, \(\pi\) is usually approximated as 3.14 or 22/7.

Keeping results in terms of \(\pi\) ensures precision, especially in theoretical calculations, while using a decimal approximation can be useful for practical situations like engineering or physics problems.