Find the length of the chord in a unit circle

Consider a unit circle with center at the origin (0,0). Let the endpoints of the chord be at coordinates (cos θ, sin θ) and (cos φ, sin φ).

The formula for finding the distance between two points (x1, y1) and (x2, y2) is given by:

$ d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $

For points (cos θ, sin θ) and (cos φ, sin φ), the distance (chord length) is:

$ d = \sqrt{(\cos φ – \cos θ)^2 + (\sin φ – \sin θ)^2} $

Using trigonometric identities, we get:

$ d = \sqrt{2 – 2 \cos(θ – φ)} $

Given that this is a unit circle, we simplify as:

$ d = 2 \sin \left(\frac{θ – φ}{2}\right) $

Therefore, the length of the chord is:

$ 2 \sin \left(\frac{θ – φ}{2}\right) $

Consider a unit circle centered at the origin with a chord connecting points A(cos θ, sin θ) and B(cos φ, sin φ).

The distance formula between two points (x1, y1) and (x2, y2) is:

$ d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $

Plugging in the coordinates of points A and B:

$ d = sqrt{(cos φ – cos θ)^2 + (sin φ – sin θ)^2} $

Using sum-to-product identities for trigonometric functions:

$ d = sqrt{2 – 2 cos(θ – φ)} $

Since the circle has a radius of 1, the chord length is:

$ d = 2 sin left(frac{θ – φ}{2}
ight) $

In a unit circle, the length of a chord joining points (cos θ, sin θ) and (cos φ, sin φ) is given by:

$ d = 2 sin left(frac{θ – φ}{2}
ight) $

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