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Find the reference angle for a rotation of 
(13 pi)/(12).
Answer:

Find the reference angle for a rotation of 13π12 \frac{13 \pi}{12} .\newlineAnswer:\newline

Full solution

Q. Find the reference angle for a rotation of 13π12 \frac{13 \pi}{12} .\newlineAnswer:\newline
  1. Understand Reference Angles: First, we need to understand what a reference angle is. A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is always between 00 and π/2\pi/2 radians (or 00 and 9090 degrees) and is found by considering the angle's location relative to the nearest x-axis.
  2. Determine Angle Location: To find the reference angle for (13π/12)(13\pi/12) radians, we need to determine where this angle lies on the unit circle. Since π\pi radians is equivalent to 180180 degrees, (13π/12)(13\pi/12) radians is more than π\pi but less than 3π/23\pi/2 radians (or 180180 degrees but less than 270270 degrees), which places it in the third quadrant.
  3. Calculate Reference Angle: In the third quadrant, the reference angle is found by subtracting the angle from π\pi (or 180180 degrees). So, we calculate the reference angle as π(13π12)\pi - (\frac{13\pi}{12}).
  4. Perform Subtraction: Perform the subtraction to find the reference angle: Reference angle = π13π12=12π1213π12=π12\pi - \frac{13\pi}{12} = \frac{12\pi}{12} - \frac{13\pi}{12} = -\frac{\pi}{12}. Since a reference angle must be positive, we take the absolute value to get π12\frac{\pi}{12} radians.
  5. Correct Mistake: However, we made a mistake in the previous step. The angle (13π/12)(13\pi/12) is more than π\pi, so we should have subtracted it from 2π2\pi to find the reference angle in the third quadrant. Let's correct this:\newlineReference angle = 2π(13π/12)=(24π/12)(13π/12)=11π/122\pi - (13\pi/12) = (24\pi/12) - (13\pi/12) = 11\pi/12 radians.

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