Problem 13 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$30^{\circ}$$

Step-by-Step Solution

Answer

The angle \(30^{\circ}\) is \(\frac{\pi}{6}\) radians.

Step 1: Understand the Conversion Relationship

To convert degrees to radians, use the formula that relates degrees to radians: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]. This formula arises because a full circle is \(360^{\circ}\) and \(2\pi\) radians.

Step 2: Apply the Conversion Formula

Plug the given angle in degrees into the conversion formula. For \(30^{\circ}\), the calculation is as follows: \[ 30 \times \frac{\pi}{180} \].

Step 3: Simplify the Expression

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. In this case, \(30\) and \(180\) share a GCD of \(30\). So, \[ \frac{30\pi}{180} = \frac{30}{180} \times \pi = \frac{1}{6} \times \pi = \frac{\pi}{6} \].

Key Concepts

Angle Conversion

Converting angles from degrees to radians allows us to work seamlessly within mathematical equations that involve circular motion or periodic functions. The conversion formula is simple yet powerful:

• Radians = Degrees \( \times \frac{\pi}{180} \)

This formula helps us understand the relationship between the degree and radian measurement systems. It originates from the fact that a full circle in degrees is \( 360^\circ \), which corresponds to \( 2\pi \) in radians.
To convert any given angle, you multiply the degree measure by \( \frac{\pi}{180} \). This efficiently transforms it into radians.
Degree measurement is commonly used in everyday scenarios, such as in navigation or geometry. However, radians become essential when analyzing mathematical functions, such as sine or cosine, where they provide a natural measure related to the circle's circumference.

Radians

Radians are a unit for measuring angles, and they are based on the radius of a circle. A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. This makes radians a natural unit of measure when dealing with circular structures. For example:

• One complete revolution around a circle is \( 2\pi \) radians.
• A quarter revolution is \( \frac{\pi}{2} \) radians.

Unlike degrees which divide a circle into 360 equal parts, radians relate directly to the physical lengths in a circle. This provides a more intuitive measure for many mathematical concepts.
When using radians, formulas involving angles often become simpler and more elegant, and they frequently appear in calculus work. Understanding radians provides deeper insight into the geometry of circles and the periodic nature of trigonometric functions.

Pi Calculation

Pi (\( \pi \)) is a crucial constant in mathematics, especially when it comes to converting degrees into radians. Pi represents the ratio of the circumference of a circle to its diameter, and it is approximately equal to 3.14159. However, in mathematics, \( \pi \) is an irrational number, meaning it has infinite decimal places without repeating.
In angle conversions, \( \pi \) is indispensable because of its intrinsic link to the circular nature of radians. The formula \( Radians = Degrees \times \frac{\pi}{180} \) utilizes \( \pi \) to change the measurement units accurately and reflects the geometric properties of circles.
When calculations involve circles, \( \pi \) allows us to express values concisely and accurately in terms of this fundamental constant. It is essential both in theoretical contexts and in practical calculations. By leaving angles in terms of \( \pi \), we maintain precision and clarity, particularly in advanced mathematics and physics.