Question

the image shows a scale drawing on part of a unit circle to illustrate the path of a ship traveling around earth to reach a port. according to the scale drawing, the ship is $\frac{sqrt{2}}{2}$ horizontal units from the port. which statement describes the location on the scale drawing? the angle from the origin the ship has traveled is $\frac{pi}{3}$. the ship is $\frac{1}{2}$ vertical units from the port. the radius of the circle is $\frac{sqrt{2}}{2}$. the angle from the origin that the ship traveled can be found using the equation $sin\theta=\frac{sqrt{2}}{1}$

Explanation:

Step1: Analyze the unit - circle concept

In a unit - circle, the radius \(r = 1\). The coordinates of a point \((x,y)\) on the unit - circle are given by \(x=\cos\theta\) and \(y = \sin\theta\), where \(\theta\) is the angle measured counter - clockwise from the positive \(x\) - axis.

Step2: Check each statement

• Statement 1: The angle from the origin the ship has traveled is \(\frac{\pi}{3}\). There is no information in the problem to suggest that the angle of travel is \(\frac{\pi}{3}\), so this statement is incorrect.
• Statement 2: The ship is \(\frac{1}{2}\) vertical units from the port. From the graph, we cannot directly conclude this.
• Statement 3: The radius of the circle is \(\frac{\sqrt{2}}{2}\). In a unit - circle, the radius \(r = 1\), not \(\frac{\sqrt{2}}{2}\), so this statement is incorrect.
• Statement 4: The angle from the origin that the ship traveled can be found using the equation \(\sin\theta=\frac{\sqrt{2}}{2}\). If we assume the coordinates of a point on the unit - circle related to the position of the ship, and we know that \(y = \sin\theta\). If the \(y\) - coordinate of the point representing the ship's position (relative to some reference) is \(\frac{\sqrt{2}}{2}\), then we can use the equation \(\sin\theta=\frac{\sqrt{2}}{2}\) to find the angle \(\theta\) measured from the origin. This statement is correct.

Answer:

The angle from the origin that the ship traveled can be found using the equation \(\sin\theta=\frac{\sqrt{2}}{2}\)