Question

use the information below to answer the next question
the following statements are made about the unit circle, where 0 is an angle drawn in standard position on the coordinate plane.
statement 1 the point ( aleft(\frac{sqrt{5}}{3}, \frac{-2}{3}
ight) ) lies on the unit circle.
statement 2 the point ( b(0.75,0.25) ) lies on the unit circle.
statement 3 for any point ( (x, y) ) on the unit circle, ( \tan \theta=\frac{y}{x} ).
statement 4 any point that lies on the unit circle can be described as ( (sin \theta, cos \theta) )

1. the statement(s) that are correct, in ascending order, above are: __________

Explanation:

Step1: Check if a point lies on the unit circle

The equation of the unit circle is \(x^{2}+y^{2}=1\).
For point \(A(\frac{\sqrt{5}}{3},\frac{- 2}{3})\), calculate \(x^{2}+y^{2}\):
\(x = \frac{\sqrt{5}}{3}\), \(y=\frac{-2}{3}\)
\(x^{2}+y^{2}=(\frac{\sqrt{5}}{3})^{2}+(\frac{-2}{3})^{2}=\frac{5}{9}+\frac{4}{9}=\frac{5 + 4}{9}=1\)

For point \(B(0.75,0.25)\), calculate \(x^{2}+y^{2}\):
\(x = 0.75=\frac{3}{4}\), \(y = 0.25=\frac{1}{4}\)
\(x^{2}+y^{2}=(\frac{3}{4})^{2}+(\frac{1}{4})^{2}=\frac{9}{16}+\frac{1}{16}=\frac{9 + 1}{16}=\frac{10}{16}=\frac{5}{8}
eq1\)

Step2: Check the formula for \(\tan\theta\)

For a point \((x,y)\) on the unit circle (\(x
eq0\)), \(\tan\theta=\frac{y}{x}\) (by the definition of the tangent function in the coordinate - plane: \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for a point \((x,y)\) on the unit circle \(x = \cos\theta\), \(y=\sin\theta\))

Step3: Check the parametric form of a point on the unit circle

The parametric form of a point on the unit circle is \((\cos\theta,\sin\theta)\), not \((\sin\theta,\cos\theta)\)

Answer:

Statement 1 and Statement 3