The unit circle is defined as the set of all points exactly one unit away from the center, typically the origin. Its normalized radius makes it a fundamental tool in trigonometry.

At 0 radians, the point on the unit circle is (1, 0), so the cosine, which represents the x-coordinate, is 1. This basic fact is one of the cornerstones of trigonometry.

The point (0,1) is located at 90 degrees, which is equivalent to π/2 radians on the unit circle. This is a standard reference position in trigonometry.

At π/2 radians, the y-coordinate on the unit circle is 1, and sine is defined as the y-coordinate. This makes the sine value equal to 1.

The standard equation for a circle centered at the origin is given by x² + y² = r². This equation represents all points (x, y) that are a distance r from the origin.

On the unit circle, sin(30°) equals 1/2, a ratio derived from the 30-60-90 right triangle. This value is one of the most commonly referenced trigonometric ratios.

At 45 degrees, both the sine and cosine of the angle are equal to √2/2 due to the symmetry of the 45°-45°-90° triangle. This is a standard trigonometric result.

Using the conversion factor where 180° equals π radians, 180° converts directly to π radians. This conversion is essential for working between the two measurement systems.

Since tan(θ) is defined as sin(θ)/cos(θ) and for 45° both sine and cosine are equal (√2/2), their ratio is 1. This is a foundational trigonometric identity.

At an angle of π radians, the point on the unit circle is located at (-1, 0). This reflects the symmetry of the circle along the horizontal axis.

For a unit circle with radius 1, the arc length is calculated as the product of the radius and the central angle in radians, which simplifies to θ. This direct relationship is a key concept in circle geometry.

Since π radians is equivalent to 180°, dividing 180° by 3 gives 60°. Converting between radians and degrees is a vital skill in trigonometry.

By subtracting 2π (or 12π/6) from 13π/6, the coterminal angle within the range 0 to 2π is π/6. This demonstrates the periodic nature of trigonometric angles.

A full circle measures 360° which is equivalent to 2π radians. This conversion is fundamental when working with angular measurements.

Both coordinates of the point are negative, indicating that it lies in Quadrant III. Recognizing the sign of the coordinates is essential for identifying positions on the unit circle.

Substitute x = 0.6 into the equation x² + y² = 1 to get y² = 1 - 0.36 = 0.64. Since the point is in Quadrant I, y is positive, so y = 0.8.

150° is in the second quadrant where cosine is negative and sine is positive. With a reference angle of 30°, cosine is -cos(30°) = -√3/2 and sine is sin(30°) = 1/2.

At π/3, the sine value is √3/2 and the cosine value is 1/2, so the tangent is (√3/2) divided by (1/2) which equals √3. This is a standard trigonometric result.

The chord length in a circle is given by the formula 2R·sin(θ/2). For a unit circle (R = 1) and a central angle of π/3, the chord length is 2 sin(π/6) = 2·(1/2) = 1.

A core theorem in circle geometry states that a tangent is perpendicular to the radius at the point of tangency. This is because if the line were not perpendicular, it would intersect the circle at more than one point, contradicting the definition of a tangent.