Problem 50 - Precalculus

Question

Convert from radians to degrees. Round your answers to the nearest hundredth of a degree. $$5 \sqrt{7}$$

Step-by-Step Solution

Answer

The radian \(5 \sqrt{7}\) is approximately 754.57 degrees.

Step 1: Understand the Conversion Formula

The conversion between radians and degrees is done using the formula: \[ ext{Degrees} = ext{Radians} imes \frac{180}{ ext{π}} \] This formula works because 180 degrees is equivalent to \( ext{π} \) radians.

Step 2: Substitute the Given Radians into the Formula

Substitute \( 5 \sqrt{7} \) (our given radian value) into the formula: \[ ext{Degrees} = 5 \sqrt{7} imes \frac{180}{ ext{π}} \] This will give us our degrees value in terms of an expression.

Step 3: Calculate the Expression

Calculate the expression by first finding \(5 \sqrt{7}\). Then multiply by \(\frac{180}{\pi}\). 1. Calculate \(\sqrt{7} \approx 2.64575131106459\). 2. Multiply it by 5 to get \(5 \cdot 2.64575131106459 \approx 13.22875655532295\). 3. Apply the conversion: \[ 13.22875655532295 imes \frac{180}{\pi} \approx 754.574426305\](This value was divided by \(3.14159265358979\) for precise calculation.)

Step 4: Round the Result

Now, round \(754.574426305\) to the nearest hundredth. The hundredth place is the second digit after the decimal, meaning we round: \[ 754.57 \] because the third digit is 4 and does not require rounding up.

Key Concepts

Radians

Radians are a way to measure angles that you'll often encounter in math and science. Unlike degrees, which are more commonly used in everyday conversations, radians come in handy, especially when dealing with circles or when performing trigonometry. A radian is based on the radius of a circle. It's defined as the angle created when you wrap the radius of a circle along its edge. This means one radian is the angle made when the arc length is equal to the radius.

The entire circle measures 2π radians, so if you convert that to degrees, you'd have the whole 360 degrees. This means:

• 1 radian is approximately 57.3 degrees.

When working with radians, recognizing and converting different dimensions is critical to understanding how you can apply them in various mathematical scenarios.

Degrees

Degrees are another unit to measure angles, and they are possibly the most familiar to you. We use degrees for everything from navigation to everyday talk about angles. A full circle is divided into 360 degrees, which gives us an easy way to break down angle sizes efficiently.

The choice of 360 stems from the ancient Babylonian system, which was based on the number 60. This long historical use makes it simpler to understand significant angles since we use terms like 180 degrees for a straight line or 90 degrees for a right angle.

• Common angles: 30°, 45°, 60°, 90°.

Understanding degrees is crucial, especially when you switch between different angle measurements in trigonometry and geometry.

Conversion Formula

Converting radians to degrees involves a straightforward formula. This formula relies on the fact that 180 degrees is the same as π radians. The relation between these two units is essential for converting exactly and accurately. The conversion formula is:

• Degrees = Radians × \(\frac{180}{\pi}\).

When using this formula, remember

• 180 degrees equals π radians,
• 1 radian equals roughly 57.3 degrees.

By multiplying the radian by this fraction, you effectively transform the measurement into degrees. It's a must-know formula for anyone diving into more complex mathematics.

Rounding

Rounding becomes essential after you convert and calculate angle measurements to get your final result in a usable form. It's particularly vital when your answer involves many decimal points, and you need a simpler number for practical purposes.

When rounding to the nearest hundredth, you focus on the second digit after the decimal. If this is your target for precision:

• Check the third digit after the decimal.
• If it's 5 or more, round the target digit up.
• If it's less than 5, the target digit remains the same.

For example, rounding 754.574426305 to the nearest hundredth gives you 754.57, because the third digit (4) doesn’t prompt an increase in the second digit. This method keeps your results clean and accurate enough for most uses.