Question

(-2/3, sqrt5/3) is a point on a unit circle. Find the cosine, cosecant, and sine of the angle.

Answer 1

Look at the picture.

`\csc\theta=(1)/(\sin\theta)=(1)/((y)/(r))=(r)/(y)`

We have the right triangle x, y and r. From the Pythagorean theorem we have:

`r^2=x^2+y^2\to r=√(x^2+y^2)`

We have the point

`\left(-(2)/(3);\ (\sqrt5)/(3)\right)`

Substitute:

`r=\sqrt{\left(-(2)/(3)\right)^2+\left((\sqrt5)/(3)\right)^2}\\\\r=\sqrt{(4)/(9)+(5)/(9)}\\\\r=\sqrt{(9)/(9)}\\\\r=1`

`\csc\theta=(1)/((\sqrt5)/(3))=(3)/(\sqrt5)=(3\cdot\sqrt5)/(\sqrt5\cdot\sqrt5)=\boxed{(3\sqrt5)/(5)}`