Question

(c) suppose the point ( p(x,y) ) is on the unit circle. find the missing coordinate.
(i) ( p(1, quad \times) )
(ii) ( p(quad \times, 1) )
(iii) ( p(-1, quad \times) )
(iv) ( p(quad \times,-1) )

Explanation:

Step1: Recall the equation of the unit circle

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

Step2: Solve for (i)

Given \(x = 1\), substitute into \(x^{2}+y^{2}=1\). We get \(1^{2}+y^{2}=1\), then \(y^{2}=1 - 1=0\), so \(y = 0\).

Step3: Solve for (ii)

Given \(y = 1\), substitute into \(x^{2}+y^{2}=1\). We get \(x^{2}+1^{2}=1\), then \(x^{2}=1 - 1=0\), so \(x = 0\).

Step4: Solve for (iii)

Given \(x=-1\), substitute into \(x^{2}+y^{2}=1\). We get \((-1)^{2}+y^{2}=1\), then \(y^{2}=1 - 1=0\), so \(y = 0\).

Step5: Solve for (iv)

Given \(y=-1\), substitute into \(x^{2}+y^{2}=1\). We get \(x^{2}+(-1)^{2}=1\), then \(x^{2}=1 - 1=0\), so \(x = 0\).

Answer:

(i) \(P(1,0)\)
(ii) \(P(0,1)\)
(iii) \(P(-1,0)\)
(iv) \(P(0,-1)\)