Radian Equivalent of 150 Degrees

Understanding how to convert between different units of angle measurement, such as degrees and radians, is a fundamental concept in trigonometry and mathematics. The question asks for the radian equivalent of an angle given in degrees, specifically 150 degrees.

Angle Unit Conversion Basics

Angles can be measured in several units, with the most common being degrees (\(\text{^\circ}\)) and radians. Here's a brief overview of these angle measurements:

• Degrees: A full circle is divided into 360 degrees. This unit is widely used in everyday applications, geometry, and navigation.
• Radians: A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. One full circle is equivalent to \(2\pi\) radians.

The core relationship between degrees and radians is crucial for conversion:

• \(180^\circ = \pi\) radians
• \(360^\circ = 2\pi\) radians

From this fundamental relationship, we can establish the conversion factors needed:

• To convert an angle from degrees to radians, multiply the degree value by the ratio \(\frac{\pi}{180^\circ}\).
• To convert an angle from radians to degrees, multiply the radian value by the ratio \(\frac{180^\circ}{\pi}\).

Converting 150 Degrees to Radians

To find the radian equivalent of 150 degrees, we will use the specific conversion formula from degrees to radians. This formula helps us express the same angle in a different unit that is often preferred in higher-level mathematics, especially calculus.

The formula for converting degrees to radians is:

$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180^\circ}$$

Now, let's substitute the given value of 150 degrees into this formula to calculate its radian equivalent:

$$\text{Radians} = 150^\circ \times \frac{\pi}{180^\circ}$$

Next, we need to simplify the fraction to get the most concise form of the radian measure:

$$\text{Radians} = \frac{150\pi}{180}$$

To simplify the fraction \(\frac{150}{180}\), we can divide both the numerator (150) and the denominator (180) by their greatest common divisor. Both numbers are divisible by 30:

• \(150 \div 30 = 5\)
• \(180 \div 30 = 6\)

So, the simplified fraction is \(\frac{5}{6}\).

Therefore, the radian equivalent of 150 degrees is:

$$\text{Radians} = \frac{5\pi}{6}$$

Summary of Radian Conversion

The conversion of 150 degrees to radians is a straightforward application of angle unit conversion. Understanding this process is fundamental for students studying trigonometry, geometry, and calculus, as radians are extensively used in these areas for expressing angles, particularly when dealing with circular motion, periodic functions, and series expansions.

The result \(\frac{5\pi}{6}\) radians is the precise radian equivalent of 150 degrees.