Problem 62 - Algebra 2

Question

Choose the radian measure that is equal to \(56^{\circ} .\) A. \(\frac{\pi}{15}\) B. \(\frac{7 \pi}{45}\) C. \(\frac{14 \pi}{45}\) D. \(\frac{\pi}{3}\)

Step-by-Step Solution

Answer

The radian measure equal to \(56^{\circ}\) is \(\frac{14\pi}{45}\), which is option C.

Step 1: Understanding Radian Conversion

To convert degrees to radians, we use the conversion factor \(\frac{\pi}{180}\). This factor is derived from the fact that \(180^{\circ}\) is equal to \(\pi\) radians.

Step 2: Set Up the Conversion

We need to convert \(56^{\circ}\) to radians. Using the conversion factor, we set up the equation: \(56^{\circ} \times \frac{\pi}{180}\).

Step 3: Perform the Multiplication

Multiply 56 by \(\frac{\pi}{180}\) to convert to radians. This yields \(\frac{56\pi}{180}\).

Step 4: Simplify the Fraction

Simplify \(\frac{56\pi}{180}\) by finding the greatest common divisor (GCD) of 56 and 180. The GCD is 4, so divide both the numerator and the denominator by 4, which results in \(\frac{14\pi}{45}\).

Step 5: Identify the Correct Answer

Compare the simplified fraction \(\frac{14\pi}{45}\) with the provided options. The correct option is C: \(\frac{14\pi}{45}\).

Key Concepts

Degree to Radian Conversion

Understanding how to convert degrees to radians is essential in trigonometry. Degrees and radians are two units used to measure angles. To make this conversion, you use a simple conversion factor based on the relationship between the full circle in degrees and radians: 360 degrees equals 2π radians. Thus, 180 degrees is equal to π radians. This relationship is simplified to the conversion factor:

• Multiply the degree value by \(\frac{\pi}{180}\).

To convert 56 degrees to radians, we multiply 56 by \(\frac{\pi}{180}\), which results in \(\frac{56\pi}{180}\). Simplifying the fraction gives us the radian measure. Remember this conversion method anytime you need to switch from degrees to radians in mathematical problems.

Trigonometry and Angles

Trigonometry plays a crucial role in understanding angles in a circle, and using radians is a preferred method in advanced mathematics due to its natural alignment with the unit circle. In trigonometry:

• Angles are often expressed in radians for calculations involving sine, cosine, and other trigonometric functions.
• The unit circle, which has a radius of 1, is divided into 2π radians.

This circle-centric approach simplifies many trigonometric identities. For instance, angle conversions and trigonometric calculations are straightforward. Knowing how to interchange between degrees and radians is thus fundamental not only for solving geometric queries but also for real-world applications involving wave patterns, oscillations, and more.

Mathematical Problem Solving

Problem solving in mathematics often involves converting between different units, like degrees and radians, to find solutions. Here are some general tips to tackle such problems:

• Understand the problem's requirements clearly, such as which unit of measurement is needed.
• Use conversion formulas accurately, checking each calculation step.
• Simplify complex fractions or equations to make solutions clearer and more manageable.

For the given problem of converting 56 degrees to radians, identifying the correct answer relies on precise multiplication and simplification. After using the conversion factor \(\frac{\pi}{180}\), simplifying \(\frac{56\pi}{180}\) to \(\frac{14\pi}{45}\) shows the importance of reducing fractions. In mathematical problem solving, these methodical steps ensure accuracy and help build a strong foundation for tackling advanced topics.