Problem 56 - Precalculus

Question

Convert from degrees to radians. Round your answers to three significant digits. $$298.7^{\circ}$$

Step-by-Step Solution

Answer

298.7° is approximately 5.21 radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula: \( ext{radians} = ext{degrees} imes \frac{ ext{π}}{180} \). This is because 180 degrees is equivalent to \(\pi\) radians.

Step 2: Plug in the Given Degree Value

Substitute the degree measure into the formula: \( 298.7^{\circ} \times \frac{\pi}{180} \).

Step 3: Perform the Multiplication

Calculate the multiplication: \( 298.7 \times \frac{\pi}{180} = 298.7 \times 0.0174533 \approx 5.21415 \).

Step 4: Round the Result to Three Significant Digits

The calculated radian value \( 5.21415 \) needs to be rounded to three significant digits: the result is \(5.21\) radians.

Key Concepts

Radian Measure

When we talk about angles in terms of radian measure, we're measuring the angle's size based on the arc length. In simple terms, a radian is an angle produced when we wrap the radius of a circle around its circumference. A full circle is made up of about 6.28318 radians, which is equivalent to 360 degrees in conventional degree measurements.

This measurement gives us a more natural way of describing angles, especially in calculus and trigonometry, because it relates directly to the size of the circle itself. It's important to understand that 1 radian in a circle of radius 1 (unit circle) equals the length of 1 on the circumference.

Key points to remember about radians:

• 1 radian ≈ 57.2958 degrees
• 2π radians = 360 degrees
• π radians = 180 degrees

Radians often simplify the calculations in mathematical formulas involving circles and periodic phenomena. By understanding radian measure, it becomes easier to transition to advanced mathematical concepts.

Significant Digits

Significant digits (or significant figures) are the numbers in a figure that convey meaningful information about its precision. They are critical in ensuring that calculations result in numbers that correctly reflect the precision of the measured or calculated values.

When rounding to significant digits, it's crucial to maintain the accuracy and reliability of the numbers involved in a calculation. For example, with the degree to radian conversion: initially calculated to be 5.21415 radians, it is rounded to 5.21 when focusing on three significant digits.

Key guidelines for significant digits:

• All non-zero numbers are considered significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal are significant.

Properly identifying significant digits is essential, especially in scientific and engineering calculations, where precision is paramount.

Conversion Formula

The conversion formula is the mathematical method used to change measurements from one unit to another. In the case of degree to radian conversion, understanding the formula allows one to seamlessly transition between these units.

The conversion formula is:\[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]This is derived from the basic relationship that 180 degrees correspond to \( \pi \) radians. So, you multiply the degree value by \( \frac{\pi}{180} \) to get the equivalent radian measure.

Here is how to effectively use the conversion formula:

• Identify the degree measure to be converted.
• Apply the formula: multiply the degree measure by \( \frac{\pi}{180} \).
• Proceed with the multiplication to find the radian measure.
• Round the result to the desired significant digits, if needed.

Understanding and utilizing the conversion formula accurately is essential for translating degrees into radians, especially needed in fields like physics and engineering.