Problem 47 - Contemporary Precalculus

Question

Convert the given radian measure to degrees. $$-5 \pi / 12$$

Step-by-Step Solution

Answer

Question: Convert the given radian measure $$-5\pi/12$$ into degrees. Answer: The converted radian measure into degrees is $$-75^{\circ}$$.

Step 1: Multiply the given radian measure by the conversion factor

Multiply the given radian measure $$-5\pi/12$$ by the conversion factor $$\frac{180}{\pi}$$ to convert it into degrees: $$-5\pi/12 * \frac{180}{\pi}$$

Step 2: Simplify the expression

Notice that the $$\pi$$ in the numerator and the denominator cancel out: $$\frac{-5}{12} * 180$$

Step 3: Perform the multiplication

Multiply the remaining numbers to get the degree measure: $$\frac{-5*180}{12}$$

Step 4: Simplify the fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD): $$\frac{-5*180}{12} = \frac{-900}{12} = -75$$ The converted radian measure into degrees is: $$-75^{\circ}$$.

Key Concepts

Unit Conversion

Unit conversion is a fundamental process in mathematics and science, essential for translating measurements from one system of units to another. This is particularly common when dealing with angles measured in radians and converting them to degrees. The key to understanding this process is recognizing that radians and degrees are just different ways of expressing the measure of an angle, akin to using kilometers or miles for distance. To convert radians to degrees, we use the conversion factor \( \frac{180}{\(\pi\)} \), where \(\pi\) is roughly 3.14159.

When working with negative radian measures, like \( -5\pi / 12 \), we apply the same conversion factor. Negative angles simply mean that the direction of the angle's rotation is clockwise, rather than counterclockwise. The conversion factor remains constant because the relationship between radians and degrees does not change with the sign of the angle. Just multiply the radian measure by \( \frac{180}{\pi} \) to get the equivalent degree measurement.

Simplifying Expressions

Simplifying expressions is a critical step in solving mathematical problems efficiently. It involves reducing complex or lengthy expressions into their simplest form without changing their value. This can be achieved through a variety of methods, such as canceling common factors, combining like terms, or applying properties of arithmetic operations.

In our exercise, simplifying comes after multiplying the radian measure by the conversion factor. Once we have \( -5\pi/12 \times \frac{180}{\pi} \), we notice that \( \pi \) appears in both the numerator and the denominator, which can be cancelled out -- a simplification step that helps in achieving a more refined expression. We then have a simple multiplication left, \( \frac{-5}{12} \times 180 \), which is much easier to work with. Simplification makes complex expressions clearer and often reveals a path to the solution that is more straightforward.

Multiplication in Mathematics

Multiplication in mathematics is one of the four elementary arithmetic operations and serves as a shortcut for repeated addition. Multiplication is associative and commutative, which means that the order in which we multiply numbers does not affect the result. In the context of our problem, after simplifying the expression, we arrive at the multiplication step: \( \frac{-5}{12} \times 180 \).

Here is where understanding multiplication shines; we simply multiply the numerator of the fraction by the whole number, effectively scaling the fraction by that whole number. In doing so, we find that \( -5 \times 180 \) equals \( -900 \) and that this product is still divided by 12. By performing the division \( \frac{-900}{12} \), we obtain our final answer of \( -75^\circ \). Grasping the multiplication of fractions and whole numbers is essential as it is a stepping stone to mastering more complex mathematical concepts. Lastly, it is important to simplify the product to its lowest terms, if possible, to achieve the clearest and most concise answer.