Question

the image shows a scale drawing on part of a unit circle to illustrate the path a plane takes around earth while flying to its destination. according to the scale drawing, the plane is \frac{sqrt{2}}{2} horizontal units from the destination. which statement describes the location on the scale drawing? the plane is \frac{sqrt{3}}{2} vertical units from the destination. the angle from the origin that the plane still needs to travel is \frac{pi}{4}. the radius of the circle is \frac{sqrt{2}}{2}. the angle from the origin that the plane needs to travel can be found using the equation cos\theta=\frac{1}{\frac{sqrt{2}}{2}}.

Explanation:

Step1: Analyze the unit - circle properties

In a unit - circle, the radius \(r = 1\). The coordinates of a point on the unit - circle are given by \((x,y)=(\cos\theta,\sin\theta)\).

Step2: Check each statement

1. The distance from the plane to the destination in the \(y\) - direction (vertical) is not \(\frac{\sqrt{3}}{2}\) as there is no indication in the right - triangle formed in the unit - circle to support this value.
2. The angle \(\theta\) from the origin that the plane still needs to travel is \(\frac{\pi}{4}\). In the right - triangle formed in the unit - circle, if we assume the hypotenuse is 1 (radius of the unit - circle), and the legs of the right - triangle are equal (since the right - triangle is an isosceles right - triangle), \(\cos\theta=\sin\theta=\frac{\sqrt{2}}{2}\) and the angle \(\theta = \frac{\pi}{4}\) radians.
3. The radius of the unit - circle is 1, not \(\frac{\sqrt{2}}{2}\).
4. The equation \(\cos\theta=\frac{1}{\frac{\sqrt{2}}{2}}=\sqrt{2}\) is incorrect because the range of the cosine function is \([- 1,1]\), and \(\sqrt{2}\gt1\).

Answer:

The angle from the origin that the plane still needs to travel is \(\frac{\pi}{4}\).