Draw the unit circle and a first-quadrant ray from the origin that makes an angle è with the positive x-axis. Let B be the point on this ray whose x-coordinate is 1, and let
A = (1, 0). Segment AB is tangent to the circle. In terms of è, find its length.
To find the length of segment AB in terms of the angle è, we can utilize the properties of the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) on a coordinate plane. It has coordinates (cos θ, sin θ) for any angle θ.
First, let's draw the unit circle. Start by drawing a circle with the origin (0, 0) as the center and a radius of 1. This represents the unit circle.
Next, draw a ray originating from the origin and passing through the point A = (1, 0). This ray makes an angle è with the positive x-axis. Remember that the first quadrant lies in the upper right portion of the unit circle.
The point B lies on this ray and has an x-coordinate of 1. We want to find the length of segment AB.
Since AB is tangent to the circle, it forms a right angle with the radius. Let C be the point where AB intersects the unit circle. As AC is perpendicular to AB, AC is a radius of the circle.
To find the length of segment AB, we need to find the length of AC. In terms of è, the x-coordinate of the point C is cos è, and the y-coordinate is sin è. Therefore, the coordinates of C are (cos è, sin è).
Using the distance formula, which is the Pythagorean theorem, the length of AC (and thus AB) is given by:
AB = sqrt((cos è - 1)^2 + sin^2 è)
Simplifying the equation further may depend on the specific value of è.