Question

Convert each angle in radians to degrees.\(-\frac{\pi}{2}\)

Short answer

-90°

Step-by-step solution

Understand the conversion factor

To convert an angle from radians to degrees, use the conversion factor: \[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]

Set up the conversion

Multiply the given angle in radians by the conversion factor: \[ -\frac{\pi}{2} \text{ radians} \times \frac{180}{\pi} \text{ degrees per radian} \]

Simplify the expression

Simplify the multiplication to find the equivalent angle in degrees. The \(\pi\) terms cancel out: \[ -\frac{\pi}{2} \times \frac{180}{\pi} = -\frac{180}{2} \]

Compute the final result

Complete the division to find the angle in degrees: \[ -\frac{180}{2} = -90° \]

Key concepts

Angle Conversion

Angle conversion is an essential mathematical skill, especially when working with trigonometry and geometry. It involves changing the measurement of an angle from one unit to another. In our case, we are converting angles from radians to degrees. This is vital because different fields and problems may prefer one unit over the other. Converting angles helps in standardizing measurements and ensuring accuracy in computations. To convert radians to degrees, we utilize a specific conversion factor, making the calculations straightforward and uniform.

Radians

Radians are a unit of angular measure used in mathematics and science. One radian is the angle created when the length of the arc is equal to the radius of the circle. This measure is derived from the properties of the circle and is dimensionless because it is the ratio of two lengths. Radians are often used in calculus and other mathematical contexts because they provide a natural and direct way of relating angles to other quantities like arc length and area. Due to the relationship with \(\pi\), radians are not always easy to visualize, making conversions necessary for practical applications.

Degrees

Degrees are a more familiar unit of measure for angles, particularly in our everyday lives. A full circle is divided into 360 degrees, making it easier to visualize and use. This unit has historical roots and is still widely used in many fields, including navigation, engineering, and even art. Unlike radians, which relate directly to the properties of the circle, degrees provide a straightforward way to divide and understand different angles. Most people find degrees more intuitive, which is why converting radians to degrees can help bridge the gap between mathematical theory and practical usage.

Mathematical Simplification

Mathematical simplification is a key part of solving problems efficiently. It involves reducing expressions to their simplest form to make computations easier and clearer. In our conversion problem, simplifying involves canceling common terms and performing basic arithmetic operations. For example, when converting \(-\frac{\pi}{2}\) radians to degrees, recognizing that \(\pi\) in the numerator and denominator cancels out is a significant simplification. This reduces the problem to basic multiplication and division. Simplification helps avoid unnecessary complexity, making the problem-solving process smoother and more error-free.