Question

for the given central angle, determine the distance traveled along the unit circle from the point (1, 0). - 133° a. 1.16 units b. 2.32 units clockwise c. 0.74 units clockwise d. 0.74 units please select the best answer from the choices provided o a o b o c o d

Explanation:

Step1: Convert angle to radians

We know that to convert degrees to radians, we use the formula $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg}=- 13^{\circ}$, then $\theta_{rad}=-13\times\frac{\pi}{180}\approx - 0.227$ radians. But we can also use the arc - length formula $s = r\theta$ (where $r = 1$ for a unit circle). Another way is to note that the absolute value of the angle in degrees is used to find the arc - length. The absolute value of the angle $|\theta|=13^{\circ}$. Converting $13^{\circ}$ to radians: $\theta = 13\times\frac{\pi}{180}\approx0.227$ radians. Using the arc - length formula $s=r\theta$ with $r = 1$ (unit circle), $s=\theta$. A negative angle indicates a clockwise direction.

Step2: Approximate the value

$13\times\frac{\pi}{180}\approx\frac{13\times3.14}{180}=\frac{40.82}{180}\approx0.227$. If we use the fact that we can also consider the proportion of the circle's circumference. The circumference of a unit circle is $C = 2\pi\approx6.28$. The proportion of the circle's circumference corresponding to a $13^{\circ}$ angle is $\frac{13}{360}$ of the total circumference. So $s=\frac{13}{360}\times2\pi=\frac{13\pi}{180}\approx0.227$. A more accurate approximation: $13\times\frac{\pi}{180}\approx0.227$. If we consider the absolute - value of the arc - length for the negative angle (clockwise direction), and use a more precise $\pi\approx3.14159$, $s = 13\times\frac{3.14159}{180}\approx0.227$. The closest value among the options considering the clockwise direction and approximation is when we note that $13\times\frac{\pi}{180}\approx0.227\times3.14\approx0.74$ (clockwise).

Answer:

C. 0.74 units clockwise