Alright, let's express and convert the given angles step-by-step.
1. Expressing Angles into Grades
a) Convert 2000 minutes to degrees:
- There are 60 minutes in one degree.
- Therefore, 2000 minutes is equal to [tex]\( \frac{2000}{60} \)[/tex] degrees.
- This simplifies to approximately [tex]\( 33.333333333333336 \)[/tex] degrees.
b) Convert 20000 seconds to degrees:
- There are 3600 seconds in one degree.
- Therefore, 20000 seconds is equal to [tex]\( \frac{20000}{3600} \)[/tex] degrees.
- This simplifies to approximately [tex]\( 5.555555555555555 \)[/tex] degrees.
c) Convert 200 minutes and 500 seconds to degrees:
- First, convert 200 minutes:
- [tex]\(\frac{200}{60} \)[/tex] degrees
- This is approximately [tex]\( 3.3333333333333335 \)[/tex] degrees.
- Then, convert 500 seconds:
- [tex]\(\frac{500}{3600} \)[/tex] degrees
- This is approximately [tex]\( 0.1388888888888889 \)[/tex] degrees.
- So, the sum is [tex]\( 3.3333333333333335 + 0.1388888888888889 \)[/tex].
d) Convert 50 degrees and 50 minutes to degrees:
- 50 degrees and 50 minutes:
- So, the first part is [tex]\( 50 \)[/tex] degrees.
- Then, [tex]\( \frac{50}{60} \)[/tex] degrees.
- This part is approximately [tex]\( 0.8333333333333334 \)[/tex] degrees.
- So, the sum is [tex]\( 50 + 0.8333333333333334 \)[/tex].
2. Expressing Angles into Radian Measure
a) Convert [tex]\( 45^{\circ} \)[/tex] to radians:
- The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex].
- Therefore, [tex]\( 45^{\circ} \)[/tex] is [tex]\( 45 \times \frac{\pi}{180} \)[/tex] radians.
- This is equal to [tex]\( 0.7853981633974483 \)[/tex] radians.
b) Convert 150 grades to radians:
- There are [tex]\( \frac{\pi}{200} \)[/tex] radians in one grade unit.
- Therefore, [tex]\( 150 \)[/tex] grades is [tex]\( 150 \times \frac{\pi}{200} \)[/tex] radians.
- This is approximately [tex]\( 2.356194490192345 \)[/tex] radians.
c) Convert [tex]\( 60^{\circ} \)[/tex] to radians:
- The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex].
- Therefore, [tex]\( 60^{\circ} \)[/tex] is [tex]\( 60 \times \frac{\pi}{180} \)[/tex] radians.
- This is equal to [tex]\( 1.0471975511965976 \)[/tex] radians.
d) Convert 758 degrees to radians:
- The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex].
- Therefore, [tex]\( 758^{\circ} \)[/tex] is [tex]\( 758 \times \frac{\pi}{180} \)[/tex] radians.
- This is approximately [tex]\( 13.229595730117017 \)[/tex] radians.
3. Expressing Angles into Degrees
a) Convert 80 grads to degrees:
- There are 0.9 degrees in one grade.
- Therefore, [tex]\( 80 \)[/tex] grades is [tex]\( 80 \times 0.9 \)[/tex] degrees.
- This is equal to [tex]\( 72.0 \)[/tex] degrees.
b) Convert [tex]\( \frac{\pi}{3} \)[/tex] radians to degrees:
- The conversion factor from radians to degrees is [tex]\( \frac{180}{\pi} \)[/tex].
- Therefore, [tex]\( \frac{\pi}{3} \)[/tex] radians is [tex]\( \frac{\pi}{3} \times \frac{180}{\pi} \)[/tex] degrees.
- This is equal to approximately [tex]\( 59.99999999999999 \)[/tex] degrees.
c) Convert 90 grades to degrees:
- There are 0.9 degrees in one grade.
- Therefore, [tex]\( 90 \)[/tex] grades is [tex]\( 90 \times 0.9 \)[/tex] degrees.
- This is equal to [tex]\( 81.0 \)[/tex] degrees.
d) Convert [tex]\( \frac{3 \pi}{4} \)[/tex] radians to degrees:
- The conversion factor from radians to degrees is [tex]\( \frac{180}{\pi} \)[/tex].
- Therefore, [tex]\( \frac{3 \pi}{4} \)[/tex] radians is [tex]\( \frac{3 \pi}{4} \times \frac{180}{\pi} \)[/tex] degrees.
- This is equal to approximately [tex]\( 135.0 \)[/tex] degrees.
