Question

What is primary angle is coterminal with the angle of \(\\{5(1 / 4) \pi\\}\) radians ?

Short answer

The primary angle coterminal with the angle of \(\frac{21}{4}\pi\) radians is \(\frac{13}{4}\pi\) radians.

Step-by-step solution

Convert the mixed number to an improper fraction

Convert the mixed number \(\frac{5}{1} + \frac{1}{4}\) to an improper fraction. \[\frac{5}{1} + \frac{1}{4} = \frac{5}{1} \cdot \frac{4}{4} + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}\] So the angle is \(\frac{21}{4}\pi\) radians.

Add or subtract multiples of \(2\pi\) to find a coterminal angle in the range of \(0\) to \(2\pi\)

Since \(2\pi\) is equivalent to a full circle, adding or subtracting multiples of \(2\pi\) will give us coterminal angles. Now, to find the primary angle that is coterminal with the given angle, we will check if adding or subtracting multiples of \(2\pi\) will result in an angle between \(0\) and \(2\pi\). \[\frac{21}{4}\pi - 2\pi \cdot\frac{4}{4} = \frac{21}{4}\pi - \frac{8}{4}\pi = \frac{13}{4}\pi\] Since the angle \(\frac{13}{4}\pi\) is in the range of \(0\) to \(2\pi\), it is the primary angle coterminal with the given angle. So, the primary angle coterminal with the angle of \(\frac{21}{4}\pi\) radians is \(\frac{13}{4}\pi\) radians.

Key concepts

Radians

Radians are a unit of measurement for angles that are used in many areas of mathematics. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. One radian is defined as the angle created by wrapping the radius of a circle around its circumference. An entire circle is equal to \(2\pi\) radians, which is approximately 6.28 radians. This makes the radian a natural choice for measuring angles, especially in trigonometry and calculus.
To convert from degrees to radians, you can use the formula:

• \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

Conversely, to convert from radians to degrees, the formula is:

• \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\)

This relationship allows you to switch back and forth between these two units of measure, depending on what is needed for the problem or context.

Mixed Fractions

A mixed fraction is a combination of a whole number and a proper fraction, such as \(5\frac{1}{4}\). Mixed fractions are often used in everyday contexts where parts are combined with whole numbers, like in measurements or recipes. To work with these numbers more easily in mathematics, it's often useful to convert them into improper fractions.
An improper fraction is one that has a numerator larger than its denominator. For example, the mixed fraction \(5\frac{1}{4}\) can be converted to the improper fraction \(\frac{21}{4}\) by multiplying the whole number by the denominator and adding the numerator, resulting in:

• \(5 \times 4 + 1 = 21\)
• Thus, \(\frac{21}{4}\)

This form is particularly useful when performing operations like multiplication or division, or when comparing the size of different fractions.

Improper Fractions

Improper fractions occur when the numerator (top number) is greater than the denominator (bottom number), as in \(\frac{21}{4}\). While they may look a bit awkward, improper fractions are extremely useful in calculations because they are easy to manipulate and work seamlessly with operations such as addition, subtraction, multiplication, and division. To convert an improper fraction back to a mixed fraction, if needed:

• Divide the numerator by the denominator to get a whole number.
• Use the remainder as the new numerator over the original denominator.

For example, with \(\frac{21}{4}\), divide 21 by 4 to get 5 with a remainder of 1, which converts it back to the mixed fraction \(5\frac{1}{4}\). Keeping track of these conversions is crucial in solving mathematical problems and often comes in handy in real-life situations as well.

Multiples of Pi

Understanding multiples of \(\pi\) is essential when working with angles in radians, especially in determining coterminal angles. The symbol \(\pi\) is approximately 3.14159 and is used to represent the ratio of the circumference of a circle to its diameter. Multiples of \(\pi\) arise frequently in angle measurements within trigonometry and other fields.When dealing with coterminal angles, it's crucial to remember that adding or subtracting \(2\pi\) (a full circle) results in an angle with the same position in the circle. This property allows mathematicians to find angles that share the same terminal side, known as coterminal angles.
For example, if you have an angle of \(\frac{21}{4}\pi\), you can find its coterminal angle by adding or subtracting full circles, or \(2\pi\), to fall within the primary interval of \([0, 2\pi]\). As demonstrated with \(\frac{21}{4}\pi - 2\pi = \frac{13}{4}\pi\), the resulting angle is equivalent in position but fits the primary range, making it easier to understand and visualize.