Answer:
[tex]\boxed{t=-\dfrac{\pi}{3}}[/tex]
[tex]\boxed{t=\dfrac{5\pi}{3}}[/tex]
[tex]\boxed{t=\dfrac{11\pi}{3}}[/tex]
[tex]\sin(t)=\boxed{-\dfrac{\sqrt{3}}{2}}[/tex]
[tex]\cot(t)=\boxed{-\dfrac{\sqrt{3}}{3}}[/tex]
[tex]\sec(t)=\boxed{2}[/tex]
Step-by-step explanation:
Given point:
[tex]\left(\dfrac{1}{2},-\dfrac{\sqrt{3}}{2}\right)[/tex]
The coordinates of the given point correspond to the angle 5π/3 radians on the unit circle. Therefore, three possible values of t (angles) are:
[tex]t = \dfrac{5\pi}{3}: \;\;\textsf{The angle that directly corresponds to the given coordinates.}[/tex]
[tex]t = \dfrac{5\pi}{3} + 2\pi =\dfrac{11\pi}{3}: \;\;\textsf{Adding one full revolution.}[/tex]
[tex]t = \dfrac{5\pi}{3} - 2\pi =-\dfrac{\pi}{3}: \;\;\textsf{Subtracting one full revolution.}[/tex]
So, the angles in order from smallest to largest are:
[tex]-\dfrac{\pi}{3},\;\; \dfrac{5\pi}{3}, \;\;\dfrac{11\pi}{3}[/tex]
On the unit circle, the coordinates are given by (cos(t), sin(t)).
Therefore, sin(t) is simply the y-coordinate of the given point:
[tex]\sin(t)=-\dfrac{\sqrt{3}}{2}[/tex]
The cotangent of an angle is the reciprocal of the tangent, so the ratio of the cosine to the sine:
[tex]\cot(t)=\dfrac{1}{\tan(t)}=\dfrac{\cos(t)}{\sin(t)}[/tex]
Therefore, cot(t) is simply the ratio of the x-coordinate and the y-coordinate of the given point:
[tex]\cot(t)=\dfrac{\dfrac{1}{2}}{-\dfrac{\sqrt{3}}{2}}=-\dfrac{1}{\sqrt{3}}=-\dfrac{\sqrt{3}}{3}[/tex]
The secant of an angle is the reciprocal of the cosine:
[tex]\sec(t)=\dfrac{1}{\cos(t)}[/tex]
Therefore:
[tex]\sec(t)=\dfrac{1}{\dfrac{1}{2}}=2[/tex]
