Question

1. $-\frac{47pi}{18}$

Explanation:

Step1: Simplify the fraction

Simplify \(\frac{47\pi}{18}\) by dividing \(47\) by \(18\) to find the number of full rotations and the remaining angle. \(47\div18 = 2\) with a remainder of \(11\) (since \(18\times2 = 36\) and \(47 - 36=11\)). So, \(-\frac{47\pi}{18}=-2\pi-\frac{11\pi}{18}\).

Step2: Use coterminal angles

Adding \(2\pi\) (a full rotation) to an angle gives a coterminal angle. So, \(-\frac{47\pi}{18}\) is coterminal with \(-\frac{11\pi}{18}\) (because \(-2\pi-\frac{11\pi}{18}+2\pi = -\frac{11\pi}{18}\)). We can also find a positive coterminal angle by adding \(2\pi\) to \(-\frac{11\pi}{18}\): \(-\frac{11\pi}{18}+2\pi=-\frac{11\pi}{18}+\frac{36\pi}{18}=\frac{25\pi}{18}\). If we want to find the reference angle, for an angle in the third quadrant (since \(\pi<\frac{25\pi}{18}<\frac{3\pi}{2}\)), the reference angle is \(\frac{25\pi}{18}-\pi=\frac{25\pi}{18}-\frac{18\pi}{18}=\frac{7\pi}{18}\). (Assuming the problem is to find coterminal or reference angle, typical in Trigonometry which is a subfield of Mathematics - Calculus or Algebra/Trigonometry)

Answer:

If finding a positive coterminal angle: \(\frac{25\pi}{18}\); If finding reference angle: \(\frac{7\pi}{18}\) (depending on the exact question, but the simplified coterminal positive angle is \(\frac{25\pi}{18}\))