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Convert the following radian measures to degrees: i) (7pi/12)c ii) (-5pi/3)c iii) 5c

Mathematics

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Step-by-step solution

12 steps

To convert angles from radians (denoted by 'c') to degrees, we use the conversion factor:

1c=(180π)∘1^c = \left(\frac{180}{\pi}\right)^\circ
  • i) Convert (7π12)c\left(\frac{7\pi}{12}\right)^c to degrees.
  1. Multiply the radian measure by the conversion factor.

    $$
    \left(\frac{7\pi}{12}\right)^c = \left(\frac{7\pi}{12} \times \frac{180}{\pi}\right)^\circ
    $$
    
  2. Cancel out π\pi and simplify the numerical values.

    $$
    = \left(\frac{7 \times 180}{12}\right)^\circ
    $$
    $$
    = (7 \times 15)^\circ
    $$
    $$
    = \boxed{\text{}105^\circ\text{}}
    $$
    
    • ii) Convert (−5π3)c\left(-\frac{5\pi}{3}\right)^c to degrees.
  3. Multiply the radian measure by the conversion factor.

    $$
    \left(-\frac{5\pi}{3}\right)^c = \left(-\frac{5\pi}{3} \times \frac{180}{\pi}\right)^\circ
    $$
    
  4. Cancel out π\pi and simplify the numerical values.

    $$
    = \left(-\frac{5 \times 180}{3}\right)^\circ
    $$
    $$
    = (-5 \times 60)^\circ
    $$
    $$
    = \boxed{\text{} -300^\circ\text{}}
    $$
    
    • iii) Convert 5c5^c to degrees.
  5. Multiply the radian measure by the conversion factor.

    $$
    5^c = \left(5 \times \frac{180}{\pi}\right)^\circ
    $$
    
  6. Simplify the expression.

    $$
    = \left(\frac{900}{\pi}\right)^\circ
    $$
    
  7. Calculate the approximate numerical value (using π≈3.14159\pi \approx 3.14159).

    $$
    = \left(\frac{900}{3.14159}\right)^\circ \approx 286.48^\circ
    $$
    $$
    = \boxed{\text{} \left(\frac{900}{\pi}\right)^\circ \approx 286.48^\circ\text{}}
    $$
    
    • iv) Convert (11π18)c\left(\frac{11\pi}{18}\right)^c to degrees.
  8. Multiply the radian measure by the conversion factor.

    $$
    \left(\frac{11\pi}{18}\right)^c = \left(\frac{11\pi}{18} \times \frac{180}{\pi}\right)^\circ
    $$
    
  9. Cancel out π\pi and simplify the numerical values.

    $$
    = \left(\frac{11 \times 180}{18}\right)^\circ
    $$
    $$
    = (11 \times 10)^\circ
    $$
    $$
    = \boxed{\text{}110^\circ\text{}}
    $$
    
    • v) Convert (−14)c\left(-\frac{1}{4}\right)^c to degrees.
  10. Multiply the radian measure by the conversion factor.

    $$
    \left(-\frac{1}{4}\right)^c = \left(-\frac{1}{4} \times \frac{180}{\pi}\right)^\circ
    $$
    
  11. Simplify the expression.

    $$
    = \left(-\frac{180}{4\pi}\right)^\circ
    $$
    $$
    = \left(-\frac{45}{\pi}\right)^\circ
    $$
    
  12. Calculate the approximate numerical value (using π≈3.14159\pi \approx 3.14159).

    $$
    = \left(-\frac{45}{3.14159}\right)^\circ \approx -14.32^\circ
    $$
    $$
    = \boxed{\text{} \left(-\frac{45}{\pi}\right)^\circ \approx -14.32^\circ\text{}}
    $$
    
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