Draw the unit circle and a first-quadrant ray from the origin that makes an angle theta 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 theta, find its length.

Isn't it just

tanØ = AB/1
tanØ = AB
Ø = arctan(AB) or tan^-1(AB)

The length AB = tanØ

Well, well, well, it looks like we have a geometric problem on our hands! Let's dive in, shall we?

To start off, we know that A is the point (1, 0) and B is the point (1, tan(theta)) on the ray. Now, let's find the length of segment AB.

Since A is the point (1, 0) and B is the point (1, tan(theta)), we can use the distance formula to find the length of AB. The distance formula is like measuring the length between two points, as if you were measuring the distance it would take you to walk from point A to point B. So, the distance between A and B is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in our coordinates:

d = sqrt((1 - 1)^2 + (tan(theta) - 0)^2)

Since (1 - 1) is 0, our equation simplifies to:

d = sqrt(tan(theta)^2)

Now, recall from your mathematical memory that tan(theta) is defined as the ratio of the opposite side to the adjacent side of a right triangle. But, here's the twist - we have a unit circle!

On a unit circle, the radius (since it's a unit circle, it's just one) is always adjacent to the angle. In this case, AB is the tangent line to the unit circle, and since AB is the adjacent side, it's equal to tan(theta).

So, the length of segment AB, in terms of theta, is simply tan(theta)!

Voila! We've cracked the geometric code. Segment AB has a length of tan(theta).

To find the length of segment AB in terms of theta, we can use trigonometry.

First, let's draw the unit circle and the first-quadrant ray from the origin, making an angle theta with the positive x-axis:

· (B)
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(O) · ─── (A)
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The point O represents the origin, A represents the point (1, 0), and B represents the point we want to find the length of.

Since segment AB is tangent to the unit circle, AB is perpendicular to the radius of the circle at point A.

The length of segment AB can be found using the sine function. The sine of angle theta is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In this case, AB is the length of the side opposite theta, and the radius of the unit circle, which is 1, is the length of the hypotenuse. Therefore, we have:

sin(theta) = AB / 1

AB = sin(theta)

So, the length of segment AB in terms of theta is sin(theta).

To find the length of segment AB in terms of theta, we need to understand the properties of a unit circle and the relationship between angles and lengths on the unit circle.

First, let's draw the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0, 0).

Next, draw a ray from the origin in the first quadrant that makes an angle theta with the positive x-axis. This ray intersects the circle at point B.

Now, let's calculate the length of segment AB. Since segment AB is tangent to the circle, it is perpendicular to the radius of the circle at point B.

From the given information, we know that point A is (1, 0) and point B has an x-coordinate of 1. Since the radius of the unit circle is 1, the x-coordinate of point B is equal to the length of segment AB. Therefore, the length of segment AB is 1.

So, in terms of theta, the length of segment AB is 1.