Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-12,\) find the radian measure of each angle whose degree measure is given. \(120^{\circ}\)

Step-by-Step Solution

Answer

The radian measure of \( 120^{\circ} \) is \( \frac{2}{3}\pi \).

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the formula: \[ R = D \times \frac{\pi}{180} \]where \( R \) is the radian measure and \( D \) is the degree measure.

Step 2: Substitute the Degree Value into the Formula

Substitute \( 120^{\circ} \) into the conversion formula:\[ R = 120 \times \frac{\pi}{180} \]

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{120}{180} \) by finding the greatest common divisor (GCD) of 120 and 180, which is 60.Therefore, \( \frac{120}{180} = \frac{120 \div 60}{180 \div 60} = \frac{2}{3} \).

Step 4: Calculate the Radian Measure

Substitute the simplified fraction back into the equation:\[ R = \frac{2}{3} \pi \]Therefore, the radian measure of \( 120^{\circ} \) is \( \frac{2}{3} \pi \).

Key Concepts

Degrees to Radians

Degrees and radians are both units for measuring angles. While degrees are more common in everyday usage, particularly in areas like geography and basic geometry, radians are often used in more advanced fields such as calculus and physics.
But why do we even need different units for angles? Well, the answer lies in the circle. Radians are based on the radius of a circle, helping simplify many mathematical calculations, especially in trigonometry. The conversion from degrees to radians is essential because it allows us to switch between these systems as needed.

• Degrees are a part of a scale where a full circle is 360°.
• Radians use the radius of a circle to explain angles, with one full circle being 2π radians.

This knowledge lets us convert 120° into radians, using the conversion formula, which we will discuss next.

Radian Measure

Understanding radian measure helps in simplifying many mathematical problems. A radian is defined based on the radius of a circle. Specifically, one radian is the angle at the circle's center when the arc length is equal to the circle's radius.
This provides a straightforward relation between the circle's geometry and its angle measure. Given the usefulness of radians in higher mathematics, it's important to grasp this concept early.

• When you wrap the radius of a circle around its edge, the angle you create is exactly 1 radian.
• Since the circumference of a circle is 2π times the radius, there are precisely 2π radians in a complete circle.

With radians, many trigonometric formulas become more intuitive and less cumbersome, thanks to the direct relationship between the arc length and the radius.

Conversion Formula

The conversion formula is the key link between degree measures and radian measures. This formula allows you to translate any angle from degrees into radians, which is particularly useful in science and engineering contexts where radians are preferred.
The formula is:
\[ R = D \times \frac{\pi}{180} \]
where \( R \) is the radian measure and \( D \) is the degree measure. Let's break this down with our example of converting 120°.

• Substitute: Insert 120 for \( D \) in the formula, so you have \[ 120 \times \frac{\pi}{180} \]
• Simplify: The fraction \( \frac{120}{180} \) simplifies to \( \frac{2}{3} \) because the greatest common divisor of 120 and 180 is 60.
• Result: Multiplying by \( \pi \) gives you the radian measure, \( \frac{2}{3} \pi \).

This method shows the practical application of converting degrees to radians, giving us a radian measure of \( \frac{2}{3} \pi \) for 120°. This process is efficient and ensures accuracy in the conversion.