Answer: IV, positive, - sec [tex]\frac{\pi}{4}[/tex], [tex]\sqrt{2}[/tex]
Step-by-step explanation:
a) Convert radians into degrees to see which quadrant it is in.
[text]\frac{\pi} {180}[\text]=[text]\frac{15\pi}{4x}[\text]
π(4x) = 180(15π)
x = [tex]\frac{180(15\pi)} {4\pi}[/tex]
x = 675°
675° - 360° = 315°, which is located in Quadrant IV.
b) The coordinate (cos θ, sin θ) for 315° is: [tex](\frac{\sqrt{2}} {2},-\frac{\sqrt{2}} {2})[/tex]
sec = [tex]\frac{1}{cos}[/tex] = [tex]\frac{2}{\sqrt{2}}[/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 least nonegative angle [tex]\frac{7\pi} {4}[/tex] from 2π: [tex]\frac{8\pi} {4}[/tex] - [tex]\frac{7\pi} {4}[/tex] = [tex]\frac{\pi} {4}[/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} {4}[/tex].
e) sec [tex]\frac{7\pi} {4}[/tex] = [tex]\frac{2}{\sqrt{2}}[/tex] = [tex]\frac{2}{\sqrt{2}}*\frac{\sqrt{2}}{\sqrt{2}}[/tex]= [tex]\frac{2\sqrt{2}}{2}[/tex] = [tex]\sqrt{2}[/tex]
