Как да изчислим Unit Circle
Какво е Unit Circle?
The unit circle is a circle with radius 1 centred at the origin. It provides the foundation for trigonometry — the sine and cosine of any angle equal the y and x coordinates of the corresponding point on the circle.
Формула
x² + y² = 1; point = (cos θ, sin θ)
- θ
- angle from positive x-axis (radians or degrees)
- x
- x-coordinate (cosine value) (dimensionless)
- y
- y-coordinate (sine value) (dimensionless)
Ръководство стъпка по стъпка
- 1For angle θ: point = (cos θ, sin θ)
- 2sin²θ + cos²θ = 1 (Pythagorean identity)
- 3tan θ = sin θ / cos θ
- 4Angles repeat every 360° (2π radians)
Worked Examples
Вход
θ = 30°
Резултат
sin = 0.5, cos = √3/2 ≈ 0.866, tan ≈ 0.577
Вход
θ = 45°
Резултат
sin = cos = 1/√2 ≈ 0.707, tan = 1
Frequently Asked Questions
Why is the unit circle important?
The unit circle extends trigonometry beyond triangles. Every angle and its trig values map directly to circle coordinates.
What does the Pythagorean identity sin²θ + cos²θ = 1 represent on the unit circle?
It states that any point (cos θ, sin θ) on the unit circle satisfies the circle equation x² + y² = 1.
How many degrees are in one full rotation on the unit circle?
360° or 2π radians. After that, angles repeat with the same sine and cosine values.
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