Question

(a) the unit circle is the circle centered at (0,0) with radius 1. (b) the equation of the unit circle is $x^{2}+y^{2}=1$. (c) suppose the point $p(x,y)$ is on the unit circle. find the missing coordinate. (i) $p(1, )$ (ii) $p( ,1)$ (iii) $p(-1, )$ (iv) $p( , - 1)$

Explanation:

Step1: Recall unit - circle equation

The equation of the unit circle is $x^{2}+y^{2}=1$.

Step2: Solve for (i)

Given $x = 1$, substitute into the equation $x^{2}+y^{2}=1$. We get $1^{2}+y^{2}=1$, which simplifies to $y^{2}=0$, so $y = 0$.

Step3: Solve for (ii)

Given $y = 1$, substitute into the equation $x^{2}+y^{2}=1$. We have $x^{2}+1^{2}=1$, which simplifies to $x^{2}=0$, so $x = 0$.

Step4: Solve for (iii)

Given $x=-1$, substitute into the equation $x^{2}+y^{2}=1$. Then $(-1)^{2}+y^{2}=1$, which simplifies to $y^{2}=0$, so $y = 0$.

Step5: Solve for (iv)

Given $y=-1$, substitute into the equation $x^{2}+y^{2}=1$. We obtain $x^{2}+(-1)^{2}=1$, which simplifies to $x^{2}=0$, so $x = 0$.

Answer:

(i) $0$
(ii) $0$
(iii) $0$
(iv) $0$