Answer:
See below for explanation.
Step-by-step explanation:
Each (x, y) point on the unit circle is equal to (cos θ, sin θ).
To find cot θ, where θ is the angle corresponding to the point (x, y) on the unit circle, we can use the formula:
[tex]\boxed{\cot \theta=\dfrac{\cos \theta}{\sin \theta}=\dfrac{x}{y}}[/tex]
[tex]\hrulefill[/tex]
If cot θ = 0, then x must be zero. (If y was zero, the value would be undefined). Therefore, we need to find the points on the unit circle where the x-coordinate (cos θ) is zero.
The points on the unit circle where x = 0 are:
The corresponding angles (in radians) at these points are:
[tex]\bullet \quad \dfrac{\pi}{2}\;\;\textsf{and}\;\;\dfrac{3\pi}{2}[/tex]
Therefore, the cotangent has the value of zero at π/2 and 3π/2.
[tex]\hrulefill[/tex]
If we divide a number by the same (but negative) number, we get -1.
Similarly, if we divide a negative number by the same (but positive) number, we get -1.
Therefore, if cot θ = -1, then the x-coordinate and y-coordinate of the points must be the same, but opposite signs.
The points on the unit circle where -x = y and x = -y are:
[tex]\bullet \quad \left(-\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2}\right)\;\; \textsf{and}\;\;\left(\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)[/tex]
The corresponding angles (in radians) at these points are:
[tex]\bullet \quad \dfrac{3\pi}{4}\;\;\textsf{and}\;\;\dfrac{7\pi}{4}[/tex]
Therefore, the cotangent has the value of -1 at 3π/4 and 7π/4.
