Answer: IV, positive, [tex]\frac{\pi} {6}[/tex], - sec [tex]\frac{\pi} {6}[/tex], [tex]\frac{2\sqrt{3}}{3} [/tex]
Step-by-step explanation:
a) Look at the Unit Circle to see that [tex]\frac{11\pi} {6}[/tex] = 330°, which is located in Quadrant IV.
b) The coordinate (cos θ, sin θ) for [tex]\frac{11\pi} {6}[/tex] is: [tex](\frac{\sqrt{3}} {2},\frac{-1}{2})[/tex]
sec = [tex]\frac{1}{cos}[/tex] = [tex]\frac{2}{\sqrt{3}} [/tex] which is positive
c) Since the given angle is in Quadrant IV, which is closest to the x-axis at 360° = 2π, the reference angle can be found by subtracting the given angle [tex]\frac{11\pi} {6}[/tex] from 2π: [tex]\frac{12\pi} {6}[/tex] - [tex]\frac{11\pi} {6}[/tex] = [tex]\frac{\pi} {6}[/tex]
d) the reference angle is below the x-axis so the given angle is equal to the negative of the reference angle: - sec [tex]\frac{\pi} {6}[/tex].
e) sec [tex]\frac{11\pi} {6}[/tex] = [tex]\frac{2}{\sqrt{3}} [/tex] = [tex]\frac{2}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}}[/tex] = [tex]\frac{2\sqrt{3}}{3} [/tex]
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Answer: [tex]\frac{18\pi}{11}[/tex], IV, [tex]\frac{4\pi} {11}[/tex]
Step-by-step explanation:
2π is one rotation. 2π = [tex]\frac{22\pi}{11}[/tex]
[tex]\frac{-26\pi}{11}[/tex] + [tex]\frac{22\pi}{11}[/tex] = [tex]\frac{-4\pi}{11}[/tex]
[tex]\frac{-4\pi}{11}[/tex] + [tex]\frac{22\pi}{11}[/tex] = [tex]\frac{18\pi}{11}[/tex]
Convert the radians into degrees to see which Quadrant it is in by setting up the proportion and cross multiplying:
[tex]\frac{\pi}{180}[/tex]= [tex]\frac{18\pi}{11x}[/tex]
π(11x) = (180)18π
x = [tex]\frac{180(18\pi}{11\pi}[/tex]
x = 295° which lies in Quadrant IV
Since the given angle is in Quadrant IV, which is closest to the x-axis at 360° = 2π, the reference angle can be found by subtracting the angle of least nonegative value[tex]\frac{18\pi} {11}[/tex] from 2π: [tex]\frac{22\pi} {11}[/tex] - [tex]\frac{18\pi} {11}[/tex] = [tex]\frac{4\pi} {11}[/tex]
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Answer: [tex]\frac{5\pi}{3}[/tex], IV, [tex]\frac{4\pi} {11}[/tex], [tex]\frac{\pi} {3}[/tex]
Step-by-step explanation:
2π is one rotation. 2π = [tex]\frac{6\pi}{3}[/tex]
[tex]\frac{-13\pi}{3}[/tex] + [tex]\frac{6\pi}{3}[/tex] = [tex]\frac{-7\pi}{3}[/tex]
[tex]\frac{-7\pi}{3}[/tex] + [tex]\frac{6\pi}{3}[/tex] = [tex]\frac{-\pi}{3}[/tex]
[tex]\frac{-\pi}{3}[/tex] + [tex]\frac{6\pi}{3}[/tex] = [tex]\frac{5\pi}{3}[/tex]
This is on the Unit Circle at 300°, which is located in Quadrant IV
Since the given angle is in Quadrant IV, which is closest to the x-axis at 360° = 2π, the reference angle can be found by subtracting the angle of least nonegative value[tex]\frac{5\pi} {3}[/tex] from 2π: [tex]\frac{6\pi} {3}[/tex] - [tex]\frac{5\pi} {3}[/tex] = [tex]\frac{\pi} {3}[/tex]
