find the coordinates terminal point of arc has lenght pi?

Question

an arc lies in position on the unit circle. find the coordinates of the terminal point of the arc if the arc has lenght  pi

Answer 1

The circumference of the unit circle is 2π so π is the length of a semicircular arc.

If the semicircle starts at (-1,0) its terminal point is (1,0).

If the semicircle starts at (0,1) its terminal point is (0,-1).

The equation of the unit circle is x²+y²=1.

The line y=ax is the equation of its diameters where a is any number. The diameter intersects the circle when:

x²+a²x²=1, x²=1/(1+a²), x=±√(1/(1+a²))

so 1/(1+a²)+y²=1, y²=1-1/(1+a²)=a²/(1+a²), y=±a/√(1+a²).

If a=1, x=y=±1/√2=±√2/2, so the arc has endpoints (√2/2,√2/2) and (-√2/2,-√2/2).

If a=-1, x=±√2/2, y=∓√2/2, endpoints are (√2/2,-√2/2) and (-√2/2,√2/2).

In general then, the endpoints are:

(√(1/(1+a²)),a/√(1+a²)) and (-√(1/(1+a²)),-a/√(1+a²)).