Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-0.48$$

Step-by-Step Solution

Answer

The corresponding angle in degrees, rounded to three decimal places, is -27.488 degrees.

Step 1: Understanding the conversion formula from radians to degrees

A radian is a unit of angular measurement. A full circle is equivalent to 2π radians. Since a full circle is also 360 degrees, this infers that π radians is equal to 180 degrees. Therefore, one radian is approximately 57.2958 degrees. The formula to convert radians \(\theta_r\) to degrees \(\theta_d\) is: \[\theta_d = \theta_r \times \left( \frac{180}{\pi} \right)\]

Step 2: Substituting the given value into the formula

Substituting the given value of \(\theta_r = -0.48\) into the formula: \[\theta_d = -0.48 \times \left( \frac{180}{\pi} \right)\]

Step 3: Calculating the angle in degrees

After performing the calculation, round the answer to three decimal places. Do not forget that if the angle in radian was negative, the angle in degrees will also be negative.

Key Concepts

Angle Measurement

Understanding angle measurement is key in various fields, from mathematics and physics to engineering and architecture. Angles are essentially a way to describe the amount of rotation between two intersecting lines or rays. Two common systems used to measure angles are degrees and radians.

• **Degrees**: This is perhaps the more familiar system to many of us. A full circle contains 360 degrees, making it easy to express angles as portions of a circle. Degrees are particularly useful in everyday life, such as in navigation and geography.
• **Radians**: This system is particularly favored in higher mathematics due to its natural properties. A full circle contains 2π radians. Radians relate directly to the arc length of a circle, which makes complex calculations often more straightforward.

Both systems have their own advantages, and often, angles need to be converted from one form to another, depending on the context or specific problem at hand.

Radian Calculation

Radians are a crucial unit in trigonometry and calculus, and understanding how they work is essential in these subjects. When dealing with radians, remember that they describe angles based on the radius of a circle.
One radian is the angle created when the arc length is equal to the radius of the circle. Simply put, if you "unwrap" the radius and lay it on the circumference, the subtended angle at the circle's center is one radian.

• **Relationship to π**: Because the circumference of a full circle is 2π times its radius, a full circle contains 2π radians. Thus, 1 radian is equivalent to an angle where approximately 57.2958 degrees is covered.
• **Flexibility in Math**: Radians simplify many mathematical equations, particularly those involving calculus, as they naturally incorporate the circle's inherent properties.

When converting radians into other units, such as degrees, understanding the positional and arc-based nature of radians is key.

Degree Conversion Formula

Converting between radians and degrees requires understanding the relationship between the two units. Let's delve into the conversion formula:

• **The Formula**: The formula to convert an angle \( \theta_r \) in radians to degrees \( \theta_d \) is given by: \[\theta_d = \theta_r \times \left( \frac{180}{\pi} \right)\]
• **How It Works**: This formula is derived from the fact that π radians equal 180 degrees. By multiplying the radian measure by \( \frac{180}{\pi} \), you effectively scale the radian measure to degrees.
• **Example**: Consider converting -0.48 radians. Plugging into the formula, \[\theta_d = -0.48 \times \left( \frac{180}{\pi} \right)\]. This results in an angle of approximately -27.459 degrees when rounded to three decimal places.

With the conversion formula, moving between radians and degrees becomes a straightforward process, allowing you to apply the most convenient unit to your problem.