Q. Find the reference angle for a rotation of 1213π.Answer:
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 0 and π/2 radians (or 0 and 90 degrees) and is found by considering the angle's location relative to the nearest x-axis.
Determine Angle Location: To find the reference angle for (13π/12) radians, we need to determine where this angle lies on the unit circle. Since π radians is equivalent to 180 degrees, (13π/12) radians is more than π but less than 3π/2 radians (or 180 degrees but less than 270 degrees), which places it in the third quadrant.
Calculate Reference Angle: In the third quadrant, the reference angle is found by subtracting the angle from π (or 180 degrees). So, we calculate the reference angle as π−(1213π).
Perform Subtraction: Perform the subtraction to find the reference angle: Reference angle = π−1213π=1212π−1213π=−12π. Since a reference angle must be positive, we take the absolute value to get 12π radians.
Correct Mistake: However, we made a mistake in the previous step. The angle (13π/12) is more than π, so we should have subtracted it from 2π to find the reference angle in the third quadrant. Let's correct this:Reference angle = 2π−(13π/12)=(24π/12)−(13π/12)=11π/12 radians.