Trigonometry — Definition of Points on the Unit Circle
Trigonometry — Definition of Points on the Unit Circle. Practice questions to deepen understanding of the definition of points on the unit circle. Online math practice with full solutions and step-by-step explanations.
Unit circle definition practice — definition, equation x²+y²=1, radians, key points, the four quadrants, equivalent angles.
- Definition of the unit circle (r=1, center at (0,0))
📐 Definition:
The unit circle is a circle whose center is at the origin and whose radius is:
The unit circle is a circle that:
• Center: \((0,0)\)
• Radius: \(r = 1\) ✓
Equation: \(x^2 + y^2 = 1\)
↻ Direction:
A positive angle is measured in the direction:
Counterclockwise (↺) ✓
Start: from the positive \(x\)-axis
Rotation: leftward (↺)
Clockwise (↻) ✗
Negative angles
📏 Radian:
One radian is the angle at which the arc length on the unit circle equals:
One radian = the angle at which the arc length = \(1\) ✓
(on the unit circle!)
Angle \(\theta\) (in radians) = arc length / radius
On unit circle: \(\theta = \frac{s}{1} = s\) ✓
🔄 Conversion:
\(180^\circ\) equals:
\(180^\circ = \pi\) radians ✓
This is the basic ratio for conversions!
• \(360^\circ = 2\pi\)
• \(90^\circ = \frac{\pi}{2}\)
• \(45^\circ = \frac{\pi}{4}\)
• \(30^\circ = \frac{\pi}{6}\)
• \(60^\circ = \frac{\pi}{3}\)
📍 Point:
The point on the unit circle at angle \(0\) is:
At angle \(\theta = 0\):
The point is \((1, 0)\) ✓
On the positive \(x\)-axis
📍 Point:
The point on the unit circle at angle \(\frac{\pi}{2}\) is:
At angle \(\theta = \frac{\pi}{2}\) (\(90^\circ\)):
The point is \((0, 1)\) ✓
On the positive \(y\)-axis
📍 Point:
The point on the unit circle at angle \(\pi\) is:
At angle \(\theta = \pi\) (\(180^\circ\)):
The point is \((-1, 0)\) ✓
On the negative \(x\)-axis
📍 Point:
The point on the unit circle at angle \(\frac{3\pi}{2}\) is:
At angle \(\theta = \frac{3\pi}{2}\) (\(270^\circ\)):
The point is \((0, -1)\) ✓
On the negative \(y\)-axis
🔢 Quadrants:
In the first quadrant (\(0\) to \(\frac{\pi}{2}\)), the coordinates are:
Quadrant II (\(\frac{\pi}{2}\) to \(\pi\)): \((-, +)\)
Quadrant III (\(\pi\) to \(\frac{3\pi}{2}\)): \((-, -)\)
Quadrant IV (\(\frac{3\pi}{2}\) to \(2\pi\)): \((+, -)\)
↔️ Equivalent angles:
The angles \(0\) and \(2\pi\) reach:
Angles differing by \(2\pi\) reach the same point ✓
Examples:
• \(0, 2\pi, 4\pi, \ldots\)
• \(\frac{\pi}{2}, \frac{5\pi}{2}, \frac{9\pi}{2}, \ldots\)
↻ Negative angle:
The point at angle \(-\frac{\pi}{2}\) is:
Angle \(-\frac{\pi}{2}\) = rotation clockwise (↻)
We arrive at \((0, -1)\) ✓
\(-\frac{\pi}{2}\) is equivalent to \(\frac{3\pi}{2}\) ✓
⭕ Circumference:
The circumference of the unit circle is:
Circumference = \(2\pi r\)
For unit circle: \(r = 1\)
Therefore: Circumference = \(2\pi \cdot 1 = 2\pi\) ✓
📏 Arc length:
On the unit circle, the arc length at angle \(\theta\) (in radians) equals:
Arc length = \(r \cdot \theta\)
For unit circle: \(r = 1\)
Therefore: Arc length = \(1 \cdot \theta = \theta\) ✓
The angle in radians = the arc length ✓
🔄 Conversion:
\(90^\circ\) equals:
\(180^\circ = \pi\)
Therefore: \(90^\circ = \frac{\pi}{2}\) ✓
(half of \(180^\circ\))
🔄 Conversion:
\(45^\circ\) equals:
\(90^\circ = \frac{\pi}{2}\)
Therefore: \(45^\circ = \frac{\pi}{4}\) ✓
(half of \(90^\circ\))
🔄 Conversion:
\(30^\circ\) equals:
\(180^\circ = \pi\)
Therefore: \(30^\circ = \frac{\pi}{6}\) ✓
(\(\frac{30}{180} = \frac{1}{6}\))
🔄 Conversion:
\(60^\circ\) equals:
\(180^\circ = \pi\)
Therefore: \(60^\circ = \frac{\pi}{3}\) ✓
(\(\frac{60}{180} = \frac{1}{3}\))
✓ Verification:
Which point lies on the unit circle?
We need: \(x^2 + y^2 = 1\)
\(\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2\)
\(= \frac{2}{4} + \frac{2}{4} = \frac{4}{4} = 1\) ✓
↔️ Equivalence:
The angle \(\frac{7\pi}{4}\) is equivalent to:
\(\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}\)
Full period minus \(\frac{\pi}{4}\)
Equivalent to \(-\frac{\pi}{4}\) ✓
📚 Summary:
Which statement is not true?
"The radius of the unit circle is \(\pi\)"
This is not true! ✗
The radius is \(1\), not \(\pi\)! ⚠️
✓ Definition:
• Center: \((0,0)\)
• Radius: \(r = 1\)
• Equation: \(x^2 + y^2 = 1\)
✓ Direction:
• Positive: counterclockwise ↺
• Negative: clockwise ↻
✓ Radian:
• Definition: arc length = \(1\)
• \(180^\circ = \pi\) radians
✓ Key points:
• \(\theta = 0\): \((1, 0)\)
• \(\theta = \frac{\pi}{2}\): \((0, 1)\)
• \(\theta = \pi\): \((-1, 0)\)
• \(\theta = \frac{3\pi}{2}\): \((0, -1)\)
✓ Quadrants:
• Quadrant I: \((+,+)\)
• Quadrant II: \((-,+)\)
• Quadrant III: \((-,-)\)
• Quadrant IV: \((+,-)\)
✓ Measures:
• Circumference: \(2\pi\)
• Arc length at angle \(\theta\): \(\theta\)