Question

convert the following radian measure to degrees. -5.92701 radians = \\(\square^{\circ}\square\\) (round to the nearest minute as needed)

Explanation:

Step1: Convert radians to degrees (formula: radians × $\frac{180^\circ}{\pi}$)

$-5.92701 \times \frac{180^\circ}{\pi}$

Step2: Calculate the degree measure (approximate $\pi \approx 3.1416$)

First, compute $-5.92701 \times \frac{180}{3.1416} \approx -5.92701 \times 57.2958 \approx -339.5^\circ$ (this is the decimal degree measure). Now, we need to convert the decimal part to minutes. The decimal part is $0.5$ (from $-339.5^\circ$), and since $1^\circ = 60'$, we multiply the decimal part by 60: $0.5 \times 60 = 30'$. So the measure in degrees and minutes is $-339^\circ 30'$. Wait, let's check the calculation more accurately. Let's do the first step more precisely: $-5.92701\times\frac{180}{\pi}$. Let's compute $\frac{180}{\pi}\approx57.29577951$. Then $-5.92701\times57.29577951\approx -5.92701\times57.2958$. Let's calculate $5\times57.2958 = 286.479$, $0.92701\times57.2958\approx0.9\times57.2958 = 51.56622$, $0.02701\times57.2958\approx1.547$. So total for the decimal part: $51.56622 + 1.547\approx53.11322$. So total degree measure: $- (286.479 + 53.11322) = -339.59222^\circ$. Now, the integer part is $-339^\circ$, the decimal part is $0.59222$. Multiply by 60 to get minutes: $0.59222\times60\approx35.5332'$. Rounding to the nearest minute, that's $36'$. Wait, maybe my initial approximation was wrong. Let's do the calculation with more precision. Let's use a calculator approach:

First, compute $-5.92701 \times 180 = -1066.8618$

Then divide by $\pi$: $-1066.8618 \div 3.1415926536 \approx -339.5922^\circ$

Now, take the absolute value for the decimal part (since we're dealing with minutes, the sign is for the degrees). The decimal part is $0.5922$. Multiply by 60: $0.5922\times60 = 35.532'$, which rounds to $36'$. So the angle is $-339^\circ 36'$? Wait, no, wait: $0.5922\times60 = 35.532$, which is 35.532 minutes, so nearest minute is 36? Wait, 0.532 is more than 0.5, so we round up. So 35.532 rounds to 36. So the degree measure is $-339^\circ 36'$? Wait, but let's check again. Wait, maybe I made a mistake in the sign. The radian measure is negative, so the degree measure is negative. The decimal part of the degree is 0.5922, so 0.5922*60 = 35.532 minutes, so 36 minutes when rounded to the nearest minute. So the conversion is:

Radians to degrees: multiply by $\frac{180}{\pi}$. So:

$-5.92701 \times \frac{180}{\pi} \approx -339.5922^\circ$

Now, convert decimal degrees to degrees and minutes:

The integer part is -339 degrees. The decimal part is 0.5922 (we take the absolute value for the decimal part since minutes are positive in measure, but the overall angle is negative). So 0.5922 * 60 = 35.532 minutes, which rounds to 36 minutes. So the angle is $-339^\circ 36'$? Wait, but let's check with a calculator. Let's use a calculator for the conversion:

Using a calculator, $-5.92701$ radians to degrees:

$5.92701 \times (180/\pi) \approx 5.92701 \times 57.2957795 \approx 339.592$ degrees, so negative, so $-339.592^\circ$. Now, 0.592 degrees * 60 minutes/degree = 35.52 minutes, which rounds to 36 minutes (since 0.52 is more than 0.5, we round up). So the answer is $-339^\circ 36'$? Wait, but maybe I made a mistake in the initial calculation. Wait, let's do the multiplication again:

$5.92701 \times 180 = 1066.8618$

$1066.8618 \div \pi \approx 1066.8618 \div 3.1415926535 \approx 339.5922$

Yes, so decimal degrees is -339.5922. Then, 0.5922 * 60 = 35.532, so 36 minutes. So the angle is $-339^\circ 36'$. Wait, but maybe the problem expects the answer in the form of degrees and minutes, so we need to present it as such.

Answer:

$-339^\circ 36'$