Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$-45^{\circ}$$

Step-by-Step Solution

Answer

The answer is \(-\frac{\pi}{4}\) radians.

Step 1: Understand the Relationship Between Degrees and Radians

To convert degrees to radians, we use the equivalence that \( 180^{\circ} \) is equal to \( \pi \) radians. This allows us to set up a ratio for conversion.

Step 2: Set Up the Conversion Formula

For conversion, the formula is given by \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \). We will use this formula to convert \(-45^{\circ}\) to radians.

Step 3: Apply the Conversion Formula

Plug \(-45\) into the formula: \( -45 \times \frac{\pi}{180} \). Calculate the multiplication to convert the angle.

Step 4: Simplify the Expression

Simplify \( \frac{-45\pi}{180} \). Divide both the numerator and the denominator by 45, which gives \( \frac{-\pi}{4} \).

Key Concepts

Angle Measurement

Angles can be measured in two primary units: degrees and radians. Degrees are most commonly used in everyday life, where a full circle is divided into 360 equal parts. Each part is called a degree, and thus there are 360 degrees in a full circle. For example, a right angle is 90 degrees. Conversely, radians are used mainly in higher-level mathematics and physics. They provide a simple relationship between the radius and the arc length of a circle. In radians, a full circle measures approximately 6.2832 radians (which is equivalent to 2π radians). Hence, understanding angle measurement in both units is crucial for converting between them seamlessly.

By understanding how angles are measured, students can better grasp the relationships and conversions between different units, leading to a more robust understanding of trigonometry and geometry.

Conversion Formula

To convert measurements between degrees and radians, a specific formula is used. This formula leverages the equivalency between 180 degrees and \( \pi \) radians.

• Conversion Formula: The formula to convert degrees (\( \, ^\circ \)) to radians is \( \, \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).
• This formula establishes a proportional relationship, making conversion straightforward. All you need is to multiply the degree measure by \( \frac{\pi}{180} \) to find its equivalent in radians.
• Conversely, to convert from radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).

Using the conversion formula not only helps in calculations but also improves one's understanding of how angles are related regardless of their units. Following the formula correctly is essential to solving such problems efficiently.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, especially when working with conversions and expressions that involve fractions. By reducing fractions to their simplest form, calculations become easier and outputs more interpretable.

When simplifying a fraction like \( \frac{-45\pi}{180} \), the numerator and the denominator can both be divided by their greatest common divisor (GCD).

• In this instance, both -45 and 180 can be divided by 45.
• Doing this, \( \frac{-45\pi}{180} \) simplifies to \( \frac{-\pi}{4} \).

Simplifying fractions ensures that answers remain clean, precise, and are easier to use in further calculations or interpretations. Mastering this skill aids in improving overall mathematical proficiency.