Question

the given angle measures 4.2 radians. a circle with a radius 5.3 cm long is centered at the angles vertex, and the terminal point is shown. note that \\( \cos ( 4.2 ) \approx - 0.4903 \\) a. since \\( \cos ( 4.2 ) \approx - 0.4903 \\), the terminal point is about radius lengths to the right of the circles vertical diameter b. thus, the terminal point is about radius lengths to the left of the circles vertical diameter c. what is the \\( x \\)-coordinate of the terminal point in \\( \mathrm { cm } \\)? d. what is the \\( y \\)-coordinate of the terminal point in \\( \mathrm { cm } \\)?

Explanation:

Step1: Recall the definition of cosine in polar coordinates

In polar coordinates, for a point \((r,\theta)\) (where \(r\) is the radius and \(\theta\) is the angle), the \(x\) - coordinate is given by \(x = r\cos\theta\) and the relationship between the horizontal displacement from the vertical diameter and \(\cos\theta\). If \(\cos\theta=a\), the displacement in terms of radius lengths is \(|a|\). Since \(\cos(4.2)\approx - 0.4903\), the terminal point is about \(| - 0.4903|=0.4903\) radius lengths to the right of the vertical diameter (because the absolute value gives the distance).

Step2: Analyze the left - right displacement

The negative sign in \(\cos(4.2)\approx - 0.4903\) indicates the direction. The formula for the displacement from the vertical diameter: if \(x = r\cos\theta\), and we want to express it in terms of radius lengths. The displacement \(d=\cos\theta\) (in radius lengths). Since \(\cos(4.2)\approx - 0.4903\), the terminal point is about \(0.4903\) radius lengths to the left of the vertical diameter (because \(\cos\theta=x/r\), and \(x = r\cos\theta\), a negative \(x\) - value means left - ward direction from the origin (center of the circle)).

Step3: Calculate the \(x\) - coordinate

We know that \(x = r\cos\theta\). Given \(r = 5.3\mathrm{cm}\) and \(\cos(4.2)\approx - 0.4903\). Substitute the values into the formula: \(x=5.3\times(- 0.4903)\).

$$x=-5.3\times0.4903=-2.6$$

Step4: Calculate the \(y\) - coordinate

We know that \(y = r\sin\theta\). First, find \(\sin(4.2)\). Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), so \(\sin\theta=\pm\sqrt{1 - \cos^{2}\theta}\). Since \(4.2\) radians (\(\pi\approx3.14\), \(4.2\) is in the third quadrant where \(\sin\theta<0\)). \(\sin(4.2)=-\sqrt{1-( - 0.4903)^{2}}=-\sqrt{1 - 0.2404}=-\sqrt{0.7596}\approx - 0.8716\). Then \(y = r\sin\theta\), substituting \(r = 5.3\mathrm{cm}\) and \(\sin(4.2)\approx - 0.8716\)

$$y=5.3\times(-0.8716)=-4.62$$

Answer:

a. \(0.4903\)
b. \(0.4903\)
c. \(-2.6\)
d. \(-4.62\)