Problem 6 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 20^{\circ} $$

Step-by-Step Solution

Answer

The angle \(20^{\circ}\) is equal to \(\pi/9\) radians or approximately \(0.35\) radians (to the nearest hundredth).

Step 1: Understand that 180 degrees is equal to \(\pi\) radians

This is a fundamental concept which is used to convert measurements between degrees and radians. Basically, \(\pi\) radians is equal to 180 degrees.

Step 2: Set up a conversion factor

To convert the measure from degrees to radians, set up a conversion factor. For each degree, there are \(\pi/180\) radians. Therefore, the conversion factor is \(\pi/180\).

Step 3: Multiply the given angle by the conversion factor

To convert degrees to radians, multiply the given angle \(20^{\circ}\) by the conversion factor. This gives: \(20 \times (\pi/180) = \pi/9\) radians.

Step 4: Express the result as a decimal

To express the result in decimal form, substitute the value of \(\pi\) (approximately 3.14159) into the result from step 3. This gives: \(\pi/9 = 3.14159/9 = 0.35\), rounded to the nearest hundredth.

Key Concepts

Degree to Radian Conversion

Converting degrees to radians involves a simple mathematical formula that ensures you can shift between wheel-angles easily. This conversion is crucial in trigonometry, especially when working with angles in various mathematical and real-world applications. Why convert, you might ask? Because radians often make advanced math easier, as they relate directly to the radius of a circle, making calculations more natural to perform. The essential takeaway here is knowing the relationship:
\[ 180^{\circ} = \pi \text{ radians} \]

• Always remember that \pi is approximately 3.14159
• The conversion factor from degrees to radians is \pi/180
• Every angle in degrees can be converted to radians with this relationship

For example, to convert 20° to radians, use the provided formula and simply multiply 20 by \(\pi/180\). That mathematical shift gives you \(\pi/9\), effectively changing the unit of the angle from degrees to radians.

Mathematical Conversion

Mathematical conversions, such as degree to radian conversion, are fundamental operations in geometry and applied mathematics. These conversions allow for the use and application of angles across different mathematical contexts, offering a consistent and universal approach. Conversion processes themselves are often based on well-defined ratios or relationships.
In this context:

• Understanding the conversion factor \(\pi/180\) is key
• Setting up proportions comparing two related quantities is a common approach
• Conversions are about ensuring measurements make sense across disciplines

By establishing these conversion formulas, you can work seamlessly between units, like degrees and radians, that measure angles differently. This capability is essential for advanced mathematics, physics, and engineering, where precise and consistent angle measurement is crucial.

Radian Measurement

Radian is the standard unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. Essentially, one radian is the angle formed when you wrap the radius of a circle around the circumference. This measurement is essential for simplifying mathematical formulas and calculations involving trigonometric functions.

• One full circle in radians equals \(2\pi\)
• Radians provide a more natural mathematical basis especially in calculus
• They tie directly into arc length where \(\text{arc length} = \text{radius} \times \text{angle in radians}\)

When working with radian measurements, you often encounter angles expressed in terms of \(\pi\). This is because \(\pi\) naturally derives from the circle's properties and its circumference to diameter relationship. Whether you're converting angles or expanding into trigonometry, radians provide a compact and elegant measurement form.

Trigonometric Concepts

Trigonometry heavily relies on angle measurements, and understanding how to convert these angles is fundamental to solving trigonometric problems. This branch of mathematics examines the relationships between the angles and sides of triangles. Radians are preferred in trigonometry as they simplify derivative forms and integration of trigonometric functions, providing deeper insights into calculus and higher math.
Trigonometric functions like sine, cosine, and tangent all depend on angle measurement:

• Radians simplify periodic functions such as \(\sin(x)\) and \(\cos(x)\)
• Functions tend to look neater and more intuitive when expressed in radians
• The unit circle, commonly used in trigonometry, naturally aligns with radian measure

Switching from degrees to radians in trigonometry isn't just about converting formulas—it's about unlocking cleaner, more effective solutions to complex mathematical questions.