Question

On the unit circle with its center at (0, 0), an arc with a central point P (1, 0) and terminal point T (3/5, -4/5) has length x. What is the value of x?

Answer 1

The length of an arc on the unit circle can be found using trigonometric ratios. In this case, the arc with central point P(1, 0) and terminal point T(3/5, -4/5) has a length of acos(3/5).

Step-by-step explanation:

The length of an arc on the unit circle is determined by the measure of the angle it subtends at the center of the circle. In this case, the central angle is determined by the coordinates of the central and terminal points of the arc. Given that the central point is P(1, 0) and the terminal point is T(3/5, -4/5), we can find the central angle by using trigonometric ratios.

Let's find the angle POC first, where O is the center of the unit circle. Using the coordinates of P(1, 0), we can find that the radius OP has a length of 1. This means that the hypotenuse OP is equal to 1. The adjacent side OC has a length of 3/5, while the opposite side PC has a length of -4/5.

Using the cosine function, we can find the measure of angle POC as acos(3/5). The length of the arc x is then determined by the formula x = (angle POC / 2π) * 2π * radius = (acos(3/5) / 2π) * 2π = acos(3/5).