Given the unit circle, find the length of the chord subtending an angle of $ heta$ at the center. Explain the steps and provide the solution.

To find the length of the chord subtending an angle $\theta$ at the center of the unit circle, we can use the formula for the chord length:

$L = 2r \sin\left(\frac{\theta}{2}\right)$

Since the radius $r$ of the unit circle is 1, the formula simplifies to:

$L = 2 \sin\left(\frac{\theta}{2}\right)$

Let’s go through the steps:

1. Consider the unit circle with the center at the origin (0,0) and radius 1.

2. The chord subtends an angle $\theta$ at the center.

3. Draw radii from the center to the endpoints of the chord, forming an isosceles triangle.

4. Drop a perpendicular from the center to the chord, bisecting the angle $\theta$ and the chord.

5. The length of the half-chord is $\sin\left(\frac{\theta}{2}\right)$, thus the full chord length is:

$L = 2 \sin\left(\frac{\theta}{2}\right)$

Therefore, the length of the chord is:

$L = 2 \sin\left(\frac{\theta}{2}\right)$

The length of the chord subtending an angle $ heta$ at the center of a unit circle can be determined using the following trigonometric approach.

Consider the unit circle centered at the origin with radius 1. The chord subtends an angle $ heta$ at the center. The formula for the length of the chord is:

$L = 2 sinleft(frac{ heta}{2}
ight)$

Steps:

1. Draw the radii to the endpoints of the chord, creating an isosceles triangle.

2. The angle between the radii is $ heta$.

3. Drop a perpendicular from the center to the chord, bisecting the angle $ heta$ and the chord.

4. The length of the half-chord is $sinleft(frac{ heta}{2}
ight)$ because of the right triangle formed.

5. Therefore, the full length of the chord is:

$L = 2 sinleft(frac{ heta}{2}
ight)$

Thus, the solution is:

$L = 2 sinleft(frac{ heta}{2}
ight)$

The length of the chord subtending an angle $ heta$ in a unit circle is given by:

$L = 2 sinleft(frac{ heta}{2}
ight)$

Explanation:

1. For a unit circle, the radius $r = 1$.

2. Chord length formula:

$L = 2 sinleft(frac{ heta}{2}
ight)$

Final chord length:

$L = 2 sinleft(frac{ heta}{2}
ight)$

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