Question

Convert each angle in degrees to radians. Express your answer as a multiple of \(\pi .\) \(-30^{\circ}\)

Short answer

-\(\frac{\pi}{6}\)

Step-by-step solution

Recall the conversion formula

To convert an angle from degrees to radians, use the conversion formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

Substitute the given angle

Substitute the given angle \(-30^{\circ}\) into the conversion formula: \[ \text{radians} = -30^{\circ} \times \frac{\pi}{180} \]

Simplify the fraction

Simplify the fraction: \[ -30 \times \frac{\pi}{180} = - \frac{30\pi}{180} \]

Reduce the fraction

Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 30: \[ -\frac{30\pi}{180} = -\frac{\pi}{6} \]

Key concepts

radian conversion

Converting angles from degrees to radians is an essential skill in mathematics, especially in trigonometry and calculus. This process helps in transitioning from one unit of measure to another, making it easier to work with angular measurements in different contexts. The radian is a unit of angular measure in the International System of Units (SI), and it measures the angle formed by taking the radius of a circle and bending it along the circle's edge. This method of measurement is often more natural and convenient when dealing with circular and periodic functions.

degree to radian formula

The key to converting degrees to radians lies in the conversion formula. This formula stems from the fact that a full circle is comprised of 360 degrees or 2π radians. Therefore, each degree is equivalent to \(\frac{2\u03c0}{360} = \frac{\u03c0}{180}\) radians. To convert an angle given in degrees to radians, you simply multiply the angle by \(\frac{\u03c0}{180}\). Here’s how you can apply it:

• Start with the angle in degrees.
• Multiply the angle by \(\frac{\u03c0}{180}\).
• Express the result as a simplest fraction, if possible.

For instance, when converting -30 degrees to radians:
\(-30^{\text{o}} \times \frac{\u03c0}{180} = -\frac{30 \u03c0}{180}\). Simplify this to obtain \(-\frac{\u03c0}{6} \). This is the angle in radians.

simplifying fractions

Simplifying fractions is a crucial step when converting degrees to radians. This makes the resulting angle easier to interpret and use. To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

By doing this, you reduce the fraction to its simplest form. For instance, in the conversion of -30 degrees to radians, we had \(-\frac{30 \u03c0}{180}\). Here, the GCD of 30 and 180 is 30. Dividing both numerator and denominator by 30, we get:
\(-\frac{30 \u03c0}{180} = -\frac{\u03c0}{6}\). This is the simplified form of the fraction, which we use to express the angle in radians. Simplifying fractions makes calculations more manageable and results cleaner.