For the given angles, we will find the quadrant that angle lies in it.
Before we begin, the limits of each quadrant is as follows:
Quadrant I: 0 < θ < π/2
Quadrant II: π/2 < θ < π
Quadrant III: π < θ < 3π/2
Quadrant IV: 3π/2 < θ < 2π
Now, we will check the angles:
The first angle: 3π/4
The angle lies between π/2 and π
So, it is in Q II
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The second angle: 57π/8
We will subtract the multiple of 2π to get the standard angle
[tex]\frac{57\pi}{8}=\frac{57\pi}{8}-3\cdot2\pi=\frac{9\pi}{8}[/tex]
The angle 9π/8 lies between π and 3π/2
So, the angle lies in Q III
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The third angle 13π/6
[tex]\frac{13\pi}{6}=\frac{13\pi}{6}-2\pi=\frac{\pi}{6}[/tex]
The angle π/6 lies between 0 and π/2
So, the angle lies in Q I
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The fourth angle (-35π/4)
We will add (2π) or a multiple of (2π) to find the positive standard angle
[tex]-\frac{35\pi}{4}=-\frac{35\pi}{4}+5\cdot2\pi=\frac{5\pi}{4}[/tex]
the angle 5π/4 lies between π and 3π/2
So, the angle lies in Q III
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The fifth angle (-5π/6)
[tex]-\frac{5\pi}{6}=-\frac{5\pi}{6}+2\pi=\frac{7\pi}{6}[/tex]
The angle 7π/6 lies between π and 3π/2
So, the angle lies in Q III
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The last angle (-5π/11)
[tex]-\frac{5\pi}{11}=-\frac{5\pi}{11}+2\pi=\frac{17\pi}{11}[/tex]
The angle 17π/11 lies between 3π/2 and 2π
So, the angle lies in Q IV
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So, the answer will be as shown in the following picture:
