Problem 118 - Calculus Volume 1

Question

For the following exercises, convert each angle in radians to degrees. $$\frac{\pi}{2} \mathrm{rad}$$

Step-by-Step Solution

Answer

90 degrees.

Step 1: Understand the Relationship

To convert an angle from radians to degrees, we use the relationship that \[ 1 ext{ radian} = \frac{180}{\pi} ext{ degrees} \] This means we need to multiply the radian measure by \( \frac{180}{\pi} \) to obtain the degree measure.

Step 2: Set Up the Conversion Calculation

We start with the angle \( \frac{\pi}{2} \text{ rad} \). To convert this to degrees, we multiply by \( \frac{180}{\pi} \):\[ \frac{\pi}{2} \times \frac{180}{\pi} \]

Step 3: Simplify the Calculation

In the expression \( \frac{\pi}{2} \times \frac{180}{\pi} \), the \( \pi \) in the numerator will cancel with the \( \pi \) in the denominator:\[ \frac{180}{2} \]

Step 4: Complete the Division

Perform the division to find the angle in degrees:\[ \frac{180}{2} = 90 \] Thus, \( \frac{\pi}{2} \text{ rad} \) is equal to 90 degrees.

Key Concepts

Radian to Degree Conversion

When converting angles from radians to degrees, you need a vital mathematical relationship: \[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]This equation forms the backbone of the conversion process. To convert an angle in radians to degrees, you simply multiply the radian value by \( \frac{180}{\pi} \). For example, consider the angle \( \frac{\pi}{2} \text{ rad} \). Applying the conversion factor, you set up the calculation:\[ \frac{\pi}{2} \times \frac{180}{\pi} \]After simplification, the \( \pi \) terms cancel out, yielding:\[ \frac{180}{2} \]Finally, compute the division:\[ 90 \text{ degrees} \]Understanding this straightforward conversion enables you to shift seamlessly between radians and degrees, a key skill in many areas of mathematics and science.

Angle Measurement

Understanding how angles are measured is crucial for interpreting geometrical and trigonometric problems.

• Radians measure angles based on the radius of a circle. An angle of 1 radian describes an arc length equal to the radius of the circle.
• Degrees are a more visual way of thinking about angles. A full circle is 360 degrees, piecing it into sections that are easier to visualize and understand in everyday contexts.

The choice between using radians and degrees often depends on the context: - Radians are preferred in calculus and higher mathematics due to their natural relationship with the unit circle. - Degrees are more common in everyday measuring scenarios and are often used in education for their simplicity and intuitive appeal. By mastering both systems, you can toggle between mathematical and practical scenarios, enhancing your problem-solving toolkit.

Mathematical Relationships

In the realm of mathematics, understanding the relationships between different units of measurement is essential. The relationship between radians and degrees showcases how mathematical concepts can be connected through conversion factors. This relationship exists because of the inherent geometry of circles.- **Circle Geometry**: A circle has 360 degrees, but 2π radians, thus forming the conversion relationship: \[ 2\pi \text{ rad} = 360 \text{ degrees} \]- **Unit Balance**: Mathematical relationships like this ensure calculations remain consistent and coherent, providing a reliable framework for solving geometric problems.These relationships are fundamental, not just between radians and degrees, but across various mathematical domains. They allow us to take a problem defined in one system and translate it into another, facilitating solutions in contexts ranging from physics to engineering. By grasping these conversion principles, students enhance their ability to apply mathematics to real-world situations effectively.