Problem 46 - Precalculus

Question

Convert from radians to degrees. Round your answers to the nearest hundredth of a degree. $$3.27$$

Step-by-Step Solution

Answer

3.27 radians is approximately 187.79 degrees.

Step 1: Understand the conversion formula

To convert radians to degrees, you need to multiply the radian value by \( \frac{180}{\pi} \). This is because 180 degrees is equivalent to \( \pi \) radians.

Step 2: Insert the given value into the formula

We have a radian measure of 3.27. Substitute this into the conversion formula: \[ 3.27 \times \frac{180}{\pi} \]

Step 3: Calculate the product

Now, perform the calculation:1. Multiply 3.27 by 180 to get 588.6.2. Divide 588.6 by \( \pi \) (approximately 3.14159) to find the degree measure:\[ \frac{588.6}{3.14159} \approx 187.789683 \]

Step 4: Round the result

Round 187.789683 to the nearest hundredth. The hundredth place is the second digit after the decimal point. The value is 187.79 (as the digit after 7 is less than 5).

Key Concepts

Conversion Formula

The conversion formula is the key that unlocks the transformation between radian and degree measurements. It connects two different ways of measuring angles through a simple mathematical operation. When you want to convert radians into degrees, all you need to do is multiply the radian measure by the fraction \( \frac{180}{\pi} \).
This fraction comes from the relationship that 180 degrees is equal to \( \pi \) radians. Since \( \pi \) is approximately 3.14159, the formula can transform the radians into a more familiar degree measure.
Using the calculator can simplify this further, though keeping the formula in mind helps understand what's happening behind the scenes. Implementing this formula correctly will ensure the conversion process goes smoothly and accurately.

Radian Measure

Radians offer an interesting way to measure angles, often used in mathematics and physics. The radian measure is based on the radius of a circle, making it very applicable for various scientific contexts. When measuring an angle in radians, you're essentially determining how much arc length there is relative to the radius.
One important thing to know: there's about \( 6.28318 \) radians (which equals \( 2\pi \)) in a full circle. That's why radian measurements can sometimes feel a bit less intuitive for those used to thinking in degrees. However, they provide a direct way to relate an angle to a circle's properties.
In the exercise at hand, we started with 3.27 radians. This number signifies the angle's size in terms of radians, which we then transformed into degrees using our conversion formula.

Degree Measure

Degree measure is perhaps the most familiar way to think about angles. Think of it as cutting a circle into 360 parts. Each part is one degree, making degrees a convenient way to measure angles in everyday situations like geometry, navigation, and many engineering fields.
To convert from radians to degrees, as our exercise demonstrated, ensures the angle becomes more understandable in this common frame of reference. From the example, converting 3.27 radians to degrees resulted in an angle measure of approximately 187.79 degrees.
Round your answer correctly to fit common usage, especially when precision is crucial. A rounded degree measure can be essential when communicating or applying in different technical or practical situations. Knowing how to convert and round appropriately enhances clarity and accuracy in reports, projects, and problem-solving scenarios.