Geometry and Trigonometry - The Unit Circle, Radians, and Degrees

The Radian Measure: A radian is defined as the angle created when the arc length of a circle is equal to its radius. Visually, imagine taking the radius of a circle and bending it along the circumference; the angle it spans is exactly $1$ radian. Because the circumference is $2\\pi r$, a full rotation of 360^\\circ is equivalent to $2\\pi$ radians.

The Unit Circle: The unit circle is a circle with a radius of $1$ unit centered at the origin $(0,0)$ on a coordinate plane. For any angle $\\theta$ in standard position, the terminal side intersects the circle at a point $P(x, y)$. Crucially, the $x$-coordinate is $\\cos\\theta$ and the $y$-coordinate is $\\sin\\theta$. Visually, as you move around the circle, the $y$-value represents the vertical 'rise' and the $x$-value represents the horizontal 'run'.

Quadrants and Signs (ASTC): The coordinate plane is divided into four quadrants. The signs of trigonometric ratios depend on the quadrant: in Quadrant I, All are positive; in Quadrant II, only Sine is positive; in Quadrant III, only Tangent is positive; and in Quadrant IV, only Cosine is positive. Visually, this follows the signs of the $x$ and $y$ coordinates on the Cartesian plane.

Reference Angles: A reference angle $\\theta'$ is the acute version of an angle, always measured between the terminal side and the $x$-axis. Visually, no matter which quadrant the angle falls in, you can draw a right-angled triangle to the $x$-axis to find the reference angle. For example, for 150^\\circ in Quadrant II, the reference angle is 180^\\circ - 150^\\circ = 30^\\circ.

Arc Length and Sector Area: When an angle $\\theta$ is measured in radians, the distance along the edge of the circle is the arc length $l = r\\theta$. The 'slice' of the circle is the sector, and its area is $A = \\frac{1}{2}r^2\\theta$. Visually, the arc length is a portion of the total circumference, and the sector area is a portion of the total area $\\pi r^2$.

Exact Values for Special Angles: Specific angles like 30^\\circ (\\frac{\\pi}{6}), 45^\\circ (\\frac{\\pi}{4}), and 60^\\circ (\\frac{\\pi}{3}) have exact coordinates on the unit circle derived from special right triangles. For example, 45^\\circ bisects the first quadrant, resulting in the coordinate $(\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{2}}{2})$, meaning \\sin(45^\\circ) = \\cos(45^\\circ) = \\frac{\\sqrt{2}}{2}.

The Pythagorean Identity: For any point on the unit circle, the coordinates $(x, y)$ satisfy $x^2 + y^2 = 1$. Substituting the trig functions gives the identity $\\cos^2\\theta + \\sin^2\\theta = 1$. Visually, this is simply the Pythagorean theorem applied to the triangle formed by the radius, the $x$-distance, and the $y$-distance.