Problem 20 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$100^{\circ}$$

Step-by-Step Solution

Answer

\(100^{\circ}\) is \(\frac{5\pi}{9}\) radians.

Step 1: Understand the Conversion Formula

The formula to convert degrees to radians is: \[ radians = degrees \times \frac{\pi}{180} \] where \( \pi \approx 3.14159 \). This formula allows us to convert from degrees to radians by multiplying the degree measure by the fraction that represents one complete circle (in radians and degrees).

Step 2: Substitute Degrees into Formula

We need to convert \(100^{\circ}\) into radians using the formula. Substitute \(100\) for degrees:\[ 100 \times \frac{\pi}{180} \]

Step 3: Simplify the Expression

Simplify the multiplication inside the formula:\[ \frac{100 \pi}{180} \]Next, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 20.

Step 4: Calculate the Simplified Fraction

Divide both 100 and 180 by 20, the greatest common divisor:\[ \frac{100 \div 20}{180 \div 20} = \frac{5}{9} \]So the expression now is:\[ \frac{5\pi}{9} \]

Step 5: Final Result

The conversion of \(100^{\circ}\) to radians is complete. The answer, in terms of \(\pi\), is \(\frac{5\pi}{9}\).

Key Concepts

Conversion Formula

Converting degrees to radians is a useful skill, especially in trigonometry and calculus. The process relies on the conversion formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] This formula works by translating degree measurements into radians through multiplication by \(\frac{\pi}{180}\). Why this fraction? In a circle, a full rotation is both 360 degrees and \(2\pi\) radians, so \(\pi\) radians is half of that, or 180 degrees. Therefore, one degree is equivalent to \(\frac{\pi}{180}\) radians. By simply multiplying the degree measurement by this fraction, we convert degrees into radians efficiently.When using this formula, remember:

• Keep \(\pi\) in the result for precision unless an approximation is needed.
• Check to make sure that you're considering the entire angle, especially when dealing with angles beyond 360 degrees.

Understanding this basic conversion enhances your ability to solve problems in which radian measures are more convenient or necessary.

Simplifying Fractions

Once you have substituted the degree value into the conversion formula, you often end up with a fraction. Simplifying fractions can make them easier to work with, especially in the context of mathematical expressions. For instance, let's say our conversion of 100 degrees gives us \(\frac{100\pi}{180}\). The goal here is to make this fraction as simple as possible. To achieve this, we find the common factors of the numerator and denominator. By dividing both parts of the fraction by their common factors, we reduce it to its simplest form. This ensures that any further calculations are more straightforward. Simplification may seem like a small step, but it greatly helps to avoid unnecessary complications later on in the problem-solving process.

• Look for the largest number that divides both the numerator and the denominator without leaving a remainder.
• Reducing fractions also helps in comparing their values easily.

Visualizing fractions in simpler terms ensures clarity and conciseness.

Greatest Common Divisor

Identifying the greatest common divisor (GCD) is a critical step in many fraction-related problems. The GCD is the biggest number that can evenly divide two or more numbers. Using the GCD to simplify fractions involves dividing the numerator and the denominator by this number. In the example of converting 100 degrees to radians, we initially have \(\frac{100\pi}{180}\). The GCD of 100 and 180 is found to be 20. By dividing both 100 and 180 by 20, we simplify the fraction to \(\frac{5\pi}{9}\).Steps to find the GCD could involve:

• Listing the factors of each number and identifying the largest shared factor.
• Using the Euclidean algorithm, which iteratively applies the principle that the GCD of two numbers also divides their difference.

Determining the GCD quickly and accurately makes mathematical simplification much more manageable. This skill is particularly beneficial for students in handling complex equations devoid of unnecessary numerals.