Question

Convert each angle in degrees to radians. Express your answer as a multiple of \(\pi .\) \(330^{\circ}\)

Short answer

The angle \( 330^{\circ} \) is \[ \frac{11\pi}{6} \] radians.

Step-by-step solution

- Understand the Conversion Factor

To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This factor relates degrees to radians.

- Set Up the Equation

Multiply the given angle in degrees by \( \frac{\pi}{180} \). For this exercise, the equation is \ 330^{\circ} \times\ \frac{\pi}{180} \.

- Simplify the Expression

Perform the multiplication: \[ 330 \times \frac{\pi}{180} = \frac{330\pi}{180} \]

- Reduce the Fraction

Simplify \( \frac{330\pi}{180} \) by finding the greatest common divisor (GCD) of 330 and 180. The GCD is 30. Divide both the numerator and the denominator by 30: \[ \frac{330/30\pi}{180/30} = \frac{11\pi}{6} \]

- State the Final Answer

The angle \( 330^{\circ} \) in radians is \[ \frac{11\pi}{6} \].

Key concepts

Angle Conversion

Angle conversion is an essential skill in mathematics, especially in trigonometry. When we talk about converting angles, we usually mean changing from one unit of measurement to another.
The two most common units are degrees and radians. While degrees are familiar and intuitive, radians offer a more natural way to describe angles mathematically.

To convert from degrees to radians, use the conversion factor: \( \frac{\text{degrees} \times \frac{\tau}{360}} \) or equivalently \( \frac{\text{degrees} \times \frac{\tau}{2}} \). The general formula is:

• Degrees to radians: \( \text{degrees} \times \frac{\tau}{360} \)
• Radians to degrees: \( \text{radians} \times \frac{360}{\tau} \)

Trigonometry

Trigonometry deals with the relationships between angles and sides in triangles, especially right-angled triangles. When working with circular functions or waveforms, radians are the preferred unit of angle.
Mentioning trigonometry helps understand why conversions between degrees and radians are so common. The trigonometric functions (sine, cosine, tangent, etc.) have simpler derivatives and integrals when expressed in radians. This makes calculations more straightforward and highlights important periodic properties.
Remember, one complete circle is 360 degrees or \(2\text {π} \) radians. Therefore, any angle can be expressed in either unit depending on the computation need.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. In this system, the angle is defined as the length of the arc created by the angle divided by the circle's radius.
This makes radians inherently tied to the circle's geometry.

To understand radians, visualize cutting a piece of string equal to the circle's radius and stretching it along the circumference. The angle formed at the circle's center by this arc equals 1 radian.
In a full circle of radius \(r\), the circumference is \(2\text {π} r\). Dividing the full circumference by \(r\), we find there are \(2\text {π}\) radians in a complete circle. This is why \(180^{\text{°}} = \text {π} \) radians, and the conversion factor \( \frac{\text {π}}{180} \) effectively connects the two units.

Fractions Simplification

When converting angles, it is often necessary to simplify fractions. Simplification involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, in the exercise, \( \frac{330\text{π}}{180} \) can be simplified. First, find the GCD of 330 and 180. Using the Euclidean algorithm or prime factorization, the GCD is 30.

Now, divide both the numerator and denominator by 30: \( \frac{330 \text{π} / 30}{180 / 30} = \frac{11 \text{π}}{6}. \)
This process ensures the fraction is in its simplest form, making it easier to understand and use in further calculations. Simplifying fractions is a fundamental skill in mathematics that helps clarity and precision in problem-solving.