Problem 2 - Algebra and Trigonometry

Question

\(1-12\) . Find the radian measure of the angle with the given degree measure. $$ 54^{\circ} $$

Step-by-Step Solution

Answer

The radian measure of \(54^{\circ}\) is \( \frac{3\pi}{10} \).

Step 1: Understand the Conversion Relation

To convert an angle from degrees to radians, recall that \(180^{\circ}\) is equivalent to \(\pi\) radians. Therefore, the conversion factor between degrees and radians is \( \frac{\pi}{180} \).

Step 2: Set Up the Conversion Expression

Given an angle of \(54^{\circ}\), we use the conversion factor to set up the expression: \(54^{\circ} \times \frac{\pi}{180}\).

Step 3: Simplify the Expression

Simplify the expression by multiplying: \( \frac{54\pi}{180} \).

Step 4: Reduce the Fraction

The fraction can be reduced by finding the greatest common divisor of 54 and 180, which is 18. This gives: \( \frac{54\pi}{180} = \frac{3\pi}{10} \).

Step 5: State the Radian Measure

The angle of \(54^{\circ}\) in radians is \( \frac{3\pi}{10} \).

Key Concepts

Degree to Radian Conversion

To understand how to convert an angle from degrees to radians, it's essential to remember that these are simply two different units for measuring angles. Just as you convert between inches and centimeters, you convert between degrees and radians.

The key to converting from degrees to radians is the conversion relation:

• Since a full circle is 360 degrees, and this same circle is also 2π radians, we can set up a ratio.
• Half a circle is 180 degrees, which equals π radians.

Using this, we come up with the conversion factor: \(\frac{\pi}{180}\). This means whenever you have an angle in degrees that you want to convert to radians, you multiply it by \(\frac{\pi}{180}\).

This factor effectively scales down the degree measure into its equivalent in radians, taking into account the difference in units.

Angle Measure

Angles are ways to express the amount of turn between two straight lines that have a common end point (the vertex).

When you work with angles, you encounter two main units:

• Degrees: This is a classic way of measuring angles and is widely used in various fields like geography and basic geometry.
• Radians: This is another unit for measuring angles, prevalent in higher-level mathematics and physics.

Radians tell you how many radii are spanned by the angle along the circle's circumference. While degrees are more intuitive, radians realize the mathematical beauty by tying the measurement directly to the circle.

Choosing the appropriate angle measure depends on the context of the problem you're solving.

Mathematical Conversion Factor

Conversion factors are crucial in mathematics for changing from one unit of measure to another, ensuring calculations remain consistent.

In the case of converting degrees to radians, the conversion factor of \(\frac{\pi}{180}\) plays a vital role. Here's why:

• It allows you to take an angle given in degrees and convert it to a radian measure.
• It hinges on the relationship between the measures of \(180^\circ\) and \(\pi\) radians, confirming consistency across conversions.

This mathematical conversion factor is simply a ratio that represents how much of a whole unit one degree or radian is. Using it properly ensures that we accurately map the full span of a circle or any angular measure you're required to convert.