Problem 30 - Precalculus

Question

Convert from radians to degrees. $$\frac{\pi}{4}$$

Step-by-Step Solution

Answer

\(\frac{\pi}{4}\) radians is 45 degrees.

Step 1: Understand the Conversion Factor

To convert an angle from radians to degrees, use the conversion factor \( 180^\pi \). This factor comes from the fact that \( (360^ = 2\pi )\).

Step 2: Set Up the Conversion Equation

Multiply the given angle in radians by the conversion factor. The equation becomes: \[\theta_{ ext{degrees}} = \theta_{ ext{radians}} imes \left(\dfrac{180^}{\pi}\right)\].

Step 3: Substitute the Given Value

Plug in the given radians value into the equation from Step 2. \[ \theta_{ ext{degrees}} = \dfrac{\pi}{4} imes \dfrac{180^}{\pi}\].

Step 4: Simplify the Expression

Cancel \(\pi\) in the numerator and denominator, and simplify the multiplication. This gives:\[ \theta_{ ext{degrees}} = \dfrac{180^}{4}\].

Step 5: Perform the Division

Carry out the division to find the angle in degrees. \[ \dfrac{180}{4} = 45\].

Key Concepts

Angle Measurement

Angles are fundamental in geometry, and they can be measured in two main units: degrees and radians. Each unit provides a different way to express the size of an angle.

• In degrees, a full rotation around a circle is 360 degrees.
• In radians, the same full rotation is expressed as \(2\pi\) radians.

While degrees might be more familiar in everyday situations, radians are often used in higher mathematics because they result in simpler and more elegant equations. To understand and work with angles effectively, it is essential to be comfortable with both measurement systems and their conversions.

Conversion Factor

To switch between measuring angles in radians and degrees, you use a special number known as the conversion factor. This factor arises from the relationship between these two units. A complete circle is \(360^\circ\), which equals \(2\pi\) radians.
Thus, the conversion factor from radians to degrees is derived from the identity:

• \(180\) degrees = \(\pi\) radians

To convert radians to degrees, you multiply the angle in radians by \(\left( \frac{180}{\pi} \right)\). This factor allows you to express the same angle in a different unit, maintaining its size but changing its representation. It's a straightforward multiplication, yet it provides an essential bridge in geometric calculations.

Degrees and Radians

Both degrees and radians serve as units for measuring angles, and each has its unique applications. Understanding their relationship and how to transition between them is crucial in various fields of study, especially calculus and trigonometry.

• Degrees are commonly used in everyday scenarios like telling time or navigating.
• Radians are more prevalent in mathematical calculations, particularly in calculus, due to the predictability of functions like sine and cosine when angles are in radians.

Switching from radians to degrees using the conversion factor \(\left( \frac{180}{\pi} \right)\) makes the math behind these calculations more approachable and interpretable in many applied settings.

Simplifying Fractions

When converting angles from radians to degrees, simplifying fractions is a necessary step to find the final answer. Given an expression like \(\frac{\pi}{4} \times \frac{180}{\pi}\), you'll notice that \(\pi\) appears in both the numerator and the denominator.

• This allows you to cancel out \(\pi\), simplifying the expression to \(\frac{180}{4}\).
• The next step is straightforward: simply divide \(180\) by \(4\) to get \(45\) degrees.

Simplifying fractions not only makes the calculations clearer but also helps eliminate potential errors that could arise from handling more complex numbers. It's a crucial skill in simplifying mathematical expressions efficiently.