Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-12,\) find the radian measure of each angle whose degree measure is given. \(240^{\circ}\)

Step-by-Step Solution

Answer

The angle \(240^{\circ}\) is \(\frac{4\pi}{3}\) radians.

Step 1: Understand the Problem

The task is to convert an angle measured in degrees to its equivalent in radians. The specific angle we need to convert is \(240^{\circ}\).

Step 2: Remember the Conversion Formula

Recall the formula to convert degrees to radians: \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\). This formula stems from the fact that \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 3: Apply the Conversion Formula

Substitute \(240\) into the conversion formula: \(240 \times \frac{\pi}{180}\).

Step 4: Simplify the Expression

First, simplify \(\frac{240}{180}\) by finding their greatest common divisor, which is 60. Thus, \(\frac{240}{180} = \frac{4}{3}\).

Step 5: Final Calculation

Multiply \(\frac{4}{3}\) by \(\pi\) to get \(\frac{4\pi}{3}\). This is the angle of \(240^{\circ}\) expressed in radians.

Key Concepts

Understanding Radian Measure

The radian is an alternative unit of measuring angles, often used in mathematical contexts such as trigonometry and calculus. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians are based on the relationship between a circle's radius and its circumference. Here, one radian is defined as the angle you get when the arc length is equal to the radius.To break it down, the circumference of a circle is given by the formula: \C = 2\pi r \where \(r\) is the radius. Therefore, a full circle in radians is \(2\pi\) radians because \(2\pi\) radians would correspond to traversing the entire circumference of the circle. As a result, one complete circle measures \(360^{\circ}\) or \(2\pi\) radians, effectively mapping \(180^{\circ}\) to \(\pi\) radians.

Degree to Radian Conversion Explained

The conversion from degrees to radians is a fundamental skill in trigonometry. Since there are \(180^{\circ}\) in \(\pi\) radians, the formula to convert degrees to radians is:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]Let's use this formula to convert a specific angle, say \(240^{\circ}\):

• Initially, plug in the value: \(240 \times \frac{\pi}{180}\).
• Next, simplify the fraction \(\frac{240}{180}\) by its greatest common divisor, which is 60. Simplifying gives \(\frac{240}{180} = \frac{4}{3}\).
• Finally, multiply the simplified fraction by \(\pi\): \(\frac{4}{3} \times \pi = \frac{4\pi}{3}\).

Thus, \(240^{\circ}\) converts to \(\frac{4\pi}{3}\) radians. This conversion process is essential for handling trigonometric functions in calculus and other mathematics disciplines.

Using Trigonometry with Radians

Trigonometry extends beyond simple geometric problems, finding applications in engineering, physics, and computer science. At its core, trigonometry deals with the relationships between the angles and sides of triangles, particularly right-angled triangles.Radians become particularly valuable in trigonometry for several reasons:

• Simplifies calculus expressions: Trigonometric functions like sine and cosine often have simpler derivatives and integrals when expressed in radians.
• Periodicity of functions: Many trigonometric functions are periodic, and their period is often expressed more naturally in radians. For example, the sine and cosine functions have a period of \(2\pi\) radians.
• Real-world modeling: Radians are used in describing harmonic motion and waves, making their calculations intuitive when modeling cyclical phenomena.

By understanding and using radians within trigonometry, students can efficiently tackle advanced mathematics topics and real-world applications.