Problem 44 - Precalculus

Question

Convert from radians to degrees. Round your answers to the nearest hundredth of a degree. 3.

Step-by-Step Solution

Answer

3 radians is approximately 171.89 degrees.

Step 1

Recall the conversion formula from radians to degrees. The formula is given by: \[ ext{Degrees} = ext{Radians} imes \frac{180}{ ext{π}}\] where π is approximately equal to 3.14159.

Step 2

Substitute the given radian value into the conversion formula. Since the given value is 3 radians, the substitution looks like this: \[ ext{Degrees} = 3 imes \frac{180}{ ext{π}}\]

Step 3

Calculate the numerical value. First, multiply 3 by 180 to get 540 and then divide by π (approximately 3.14159): \[ ext{Degrees} = \frac{540}{3.14159} \] This produces approximately 171.887 degrees.

Step 4

Round the answer to the nearest hundredth. The result from Step 3 is 171.887 degrees. When rounded to the nearest hundredth, it becomes 171.89 degrees.

Key Concepts

Understanding Radians

Radians are a way of measuring angles using the radius of a circle. Imagine you have a circle, and the radius is the line from the center to the edge. If you take that radius and lay it along the circumference, the angle that this length subtends at the center is one radian. This makes radians very useful in mathematics because they provide a natural way of relating the size of an angle to the size of a circle.

• Radians can sometimes be harder to grasp than degrees initially because they are not based on the number of parts of a full circle (like 360 degrees).
• Radians are more deeply connected with the unit circle concept, often used in trigonometry.

In fact, a full circle has exactly 2π radians, equivalent to 360 degrees. This ensures that 1 radian rounds up to about 57.2958 degrees. Knowing this, we can easily move between radians and degrees, which play a crucial role in mathematics and physics.

Defining Degrees

Degrees are a traditional unit for measuring angles and are preferred in many day-to-day settings. The whole circle is divided into 360 equal parts, with each part being one degree. Why 360? This number is largely historical and is thought to date back to ancient civilizations.

• Degrees are intuitive as they break down a circle into easy-to-understand segments (think of dividing a pie into slices).
• They are commonly used in navigation, geography, and even in determining your latitude and longitude.

One benefit of using degrees is their simplicity in splitting and adding or subtracting angles. They are easy to visualize which is why they are often used in educational settings to teach geometry effectively. Degrees provide an approachable means to start exploring more complex mathematical systems.

The Conversion Formula

To switch between radians and degrees, we use a simple conversion formula. Knowing that a full circle is 360 degrees or 2π radians, the conversion formula from radians to degrees is:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Breaking down this formula:

• Radians: The initial angle measurement you have. In our exercise, this was 3 radians.
• Multiply by 180: Because there are 180 degrees in half a circle (π radians).
• Divide by π: This converts the multiplication factor to work with radians.

Thanks to this formula, you can take any radian value and easily convert it into degrees. In practice, you simply plug in the radian value, perform the multiplication and division, and you will have your answer in degrees.Once calculated, you may need to round the result for precision, like rounding to the nearest hundredth degree as done in our exercise. Understanding and applying this conversion formula empowers you to seamlessly transition between these two units of angular measurement.