What is the length of the chord a unit circle which subtends an angle 2θ at the centre, where θ < 45°?
2 sin θ
Understanding the Unit Circle and Chord Length
Let's find the length of a chord in a unit circle that subtends a specific angle at the center. A unit circle is a circle with a radius of 1 unit.
Consider a unit circle with center O. Let AB be the chord whose length we need to find. The question states that this chord AB subtends an angle of \(2\theta\) at the center O. So, \(\angle AOB = 2\theta\).
Applying Geometry and Trigonometry
To find the length of the chord AB, we can draw a line segment from the center O perpendicular to the chord AB. Let this line segment meet the chord at point M.
- The line segment OM is perpendicular to AB (\(OM \perp AB\)).
- A line from the center perpendicular to a chord bisects the chord. This means AM = MB.
- Also, this line OM bisects the angle subtended by the chord at the center. So, \(\angle AOM = \angle BOM = \frac{1}{2} \angle AOB = \frac{1}{2}(2\theta) = \theta\).
Now consider the triangle OMA. This is a right-angled triangle because \(OM \perp AB\) at M, so \(\angle OMA = 90^\circ\).
- OA is the radius of the unit circle, so OA = 1. OA is the hypotenuse in the right triangle OMA.
- AM is half the length of the chord AB. AM is the side opposite the angle \(\theta\) in the right triangle OMA.
Using trigonometry in the right-angled triangle OMA, we can relate the angle \(\theta\), the opposite side AM, and the hypotenuse OA:
\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AM}{OA}\)
Substitute the known values OA = 1 and the angle \(\angle AOM = \theta\):
\(\sin(\theta) = \frac{AM}{1}\)
This gives us the length of AM:
\(AM = \sin(\theta)\)
Since AM is half the length of the chord AB, the total length of the chord AB is \(2 \times AM\).
\(\text{Chord Length AB} = 2 \times AM = 2 \sin(\theta)\)
The condition \(\theta < 45^\circ\) implies \(2\theta < 90^\circ\), meaning the angle subtended at the center is acute. This condition doesn't change the formula for the chord length, which holds for any central angle \(2\theta \le 180^\circ\).
Conclusion on Chord Length Calculation
The length of the chord in a unit circle that subtends an angle \(2\theta\) at the center is \(2 \sin(\theta)\).
Revision Table: Key Concepts for Chord Length
| Concept | Description | Relevance to Problem |
|---|---|---|
| Unit Circle | A circle with radius 1. | The radius (OA, OB) is 1, simplifying calculations. |
| Chord | A line segment connecting two points on the circle. | AB is the chord whose length is sought. |
| Subtended Angle | The angle formed at the center by the radii to the chord endpoints. | Given as \(2\theta\). This angle is key to the calculation. |
| Perpendicular Bisector | A line from the center perpendicular to a chord bisects the chord and the central angle. | Creates a right triangle and allows using angle \(\theta\). |
| Trigonometry (Sine) | Relates angles and side lengths in right triangles. | Used to find half the chord length (AM) using the angle \(\theta\) and radius (OA=1). |
Additional Information: General Chord Length Formula
For a circle with any radius R, the length of a chord that subtends an angle \(\alpha\) at the center is given by the formula:
\(\text{Chord Length} = 2R \sin\left(\frac{\alpha}{2}\right)\)
In this specific problem:
- The circle is a unit circle, so Radius \(R = 1\).
- The angle subtended at the center is \(\alpha = 2\theta\).
- Substituting these values into the general formula:
\(\text{Chord Length} = 2 \times 1 \times \sin\left(\frac{2\theta}{2}\right) = 2 \sin(\theta)\)
This confirms the result derived using the step-by-step geometric approach.
Understanding how to apply trigonometry to circle properties is fundamental for solving geometry problems involving chords, radii, and angles.
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