Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Round to the nearest hundredth. $$74^{\circ}$$

Step-by-Step Solution

Answer

74° is approximately 1.29 radians.

Step 1: Understand the Conversion Formula

To convert degrees to radians, use the formula: \( ext{radians} = ext{degrees} \times \frac{ ext{π}}{180} \). This formula stems from the fact that 180 degrees equals \( ext{π} \) radians.

Step 2: Substitute the Degree Value

Substitute \( 74^{ ext{°}} \) into the conversion formula: \( ext{radians} = 74 \times \frac{ ext{π}}{180} \).

Step 3: Simplify the Expression

Perform the multiplication: \( ext{radians} = 74 \times \frac{3.14159}{180} \), which simplifies to \( ext{radians} = \frac{74 \times 3.14159}{180} \).

Step 4: Calculate and Round the Result

Calculate the expression \( \frac{74 \times 3.14159}{180} \) using a calculator, which results in approximately 1.29154. Round this result to the nearest hundredth to get 1.29 radians.

Key Concepts

Conversion Formula

When converting degrees to radians, we rely on a conversion formula. This formula is straightforward but crucial in mathematics. It's given as:

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

Breaking it down, the formula uses the ratio of degrees to radians. While 180 degrees corresponds to \( \pi \) radians, this formula quickly helps us translate any degree measure into its radian counterpart by the scaling factor of \( \frac{\pi}{180} \). Make sure to memorize or note down this relationship as it's frequently used.

Radian

A radian is a unit of angular measure used in many areas of mathematics. Unlike degrees, radians provide a natural way to describe angles using the properties of circles. One way to visualize a radian is to think of it as the angle made when you take the radius of a circle and wrap it along the circle's circumference.

• 1 full circle = 2\( \pi \) radians
• 1 radian is approximately 57.3 degrees

Radians are often preferred in mathematical calculations, thanks to their direct relationship with the properties of a circle. While degrees might be more intuitive at first, radians enable more elegant mathematical expressions and simplifications.

Degrees

Degrees are another way to measure angles. They are perhaps the most commonly understood metric thanks to their use in everyday contexts, like navigation and time reading. A full circle is divided into 360 equal parts, known as degrees. Here's what you need to know:

• A straight angle measures 180 degrees.
• 90 degrees make a right angle.
• 360 degrees complete a circle.

Degrees offer an intuitive way to convey angle measures in daily life, and converting them to radians can yield more precise mathematical computations.

Mathematical Rounding

Mathematical rounding helps simplify numbers, making them easier to understand and work with. In the context of converting degrees to radians, rounding is the final step. It involves trimming the decimal places to a desired precision, usually for practical purposes.

• Rounding to the nearest hundredth removes all but two decimal places.
• If the third decimal place is 5 or higher, round the second decimal place up.
• If the third decimal place is less than 5, keep the second decimal place as is.

In exercises involving degree to radian conversion, you'll often round your final answer to make it manageable and fit within typical precision requirements.