Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-12,\) find the radian measure of each angle whose degree measure is given. \(160^{\circ}\)

Step-by-Step Solution

Answer

The radian measure of \(160^{\circ}\) is \(\frac{8\pi}{9}\).

Step 1: Understanding Degree to Radian Conversion

To convert an angle from degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This means that to convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).

Step 2: Apply Conversion Formula

Take the degree measure \( 160^{\circ} \) and multiply it by the conversion factor. So, the expression becomes \( 160 \times \frac{\pi}{180} \).

Step 3: Simplify the Expression

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. So, \( 160 \times \frac{\pi}{180} = \frac{160}{180} \times \pi = \frac{8}{9} \times \pi \).

Step 4: Write the Result

The simplified radian measure of the angle is \( \frac{8\pi}{9} \). Hence, the conversion is complete.

Key Concepts

Converting Angles: Degrees to Radians

Angle conversion is a common task in mathematics that involves transforming an angle from one unit of measurement to another. We often need to switch between degrees and radians, especially in trigonometry and calculus. Understanding the degree to radian conversion process is essential for many areas of mathematics and engineering.

To convert degrees to radians, we use the ratio between the circle's degrees and radians. One complete revolution of a circle is 360 degrees, which is equivalent to a radian measure of \(2\pi\). Therefore, one degree is equal to \(\frac{\pi}{180}\) radians. When you have an angle in degrees, you simply multiply that angle by \(\frac{\pi}{180}\) to convert it to radians.

Here’s how it works in action:

• Start with your angle in degrees, such as 160°.
• Multiply the degree measure by \(\frac{\pi}{180}\).
• Simplify the resulting expression if possible.

This straightforward multiplication ensures an accurate conversion from degrees to radians every time.

Understanding Radian Measure

Radian measure is an alternative way of expressing angles that is often more useful in higher mathematics due to its natural relationship with the circle. A radian is defined based on the radius of a circle. Specifically, an angle has a radian measure of 1 when it subtends an arc of the circle equal in length to the circle's radius.

In the radian system, angles are expressed in terms of \(\pi\), making calculations simpler, especially in trigonometry. One full rotation around a circle covers \(2\pi\) radians. Hence, common angle measures become simpler to handle, with \(\pi\) representing half a circle (180°), and \(2\pi\) a full circle (360°).

Key points to remember:

• Radian is the standard unit for angular measure used in science and engineering.
• Common conversions include: 180° = \(\pi\) radians, 90° = \(\pi/2\) radians.
• Radian measures allow for easier application of calculus techniques.

For everyday purposes like time-telling or geographic directions, degrees are often more intuitive, but radians offer a powerful tool for mathematical analysis.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, ensuring that expressions are presented in their simplest form. When converting degrees to radians, we often deal with fractional expressions that need simplification.

In the context of degree to radian conversion, this usually involves taking the fraction obtained from multiplying the degree by the conversion factor \(\frac{\pi}{180}\) and simplifying it.

Here’s how you simplify the fraction specifically:

• Write the multiplication result as a fraction. For example, \(160 \times \frac{\pi}{180} = \frac{160\pi}{180}\).
• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD to simplify.

Using the example of converting 160°:

• The GCD of 160 and 180 is 20.
• Dividing them both by 20 gives \(\frac{8\pi}{9}\).

Once simplified, the fraction reflects the angle's simplest radian form, making further calculations more manageable and error-free.