Geometry and Trigonometry - The Unit Circle and Radian Measure

Radian Measure: A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Visually, imagine taking the radius of a circle and wrapping it along the edge; the angle formed is $1$ radian. Because the total circumference is $2\pi r$, a full rotation of $360^\circ$ is equal to $2\pi$ radians.

The Unit Circle: This is a circle with a radius of $r = 1$ centered at the origin $(0,0)$ in the Cartesian plane. For any point $P(x, y)$ on the circle making an angle $\theta$ with the positive x-axis, the coordinates are defined as $x = \cos \theta$ and $y = \sin \theta$. Visually, the cosine represents the horizontal displacement from the y-axis, and the sine represents the vertical displacement from the x-axis.

Arc Length and Sector Area: When an angle $\theta$ is measured in radians, the calculations for circles are simplified. The arc length $s$ is the distance along the curved edge of the sector, and the sector area $A$ is the space enclosed by the two radii and the arc. Visually, a sector is like a slice of pie where the crust length is the arc length.

Quadrant Signs (ASTC): The signs of trigonometric ratios depend on which quadrant the terminal side of the angle $\theta$ lies in. In Quadrant I (top-right), all ratios are positive. In Quadrant II (top-left), only $\sin \theta$ is positive. In Quadrant III (bottom-left), only $\tan \theta$ is positive. In Quadrant IV (bottom-right), only $\cos \theta$ is positive. This 'All Students Take Calculus' rule follows from the positive/negative signs of $x$ and $y$ coordinates on the unit circle.

Exact Values and Reference Angles: For any angle $\theta$, the reference angle $\theta_{ref}$ is the acute angle formed between the terminal side and the x-axis. Visually, if an angle is in the second quadrant, its reference angle is $\pi - \theta$. This allows us to use exact values from the first quadrant (like $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$) to find values for larger angles by applying the appropriate quadrant sign.

The Pythagorean Identity: For any point on the unit circle, the relationship between sine and cosine is governed by the equation of the circle $x^2 + y^2 = 1$. Substituting the trig ratios gives the fundamental identity $\cos^2 \theta + \sin^2 \theta = 1$. Visually, this is simply the Pythagorean theorem applied to a right-angled triangle with hypotenuse $1$ inside the unit circle.

Tangent Ratio: On the unit circle, $\tan \theta$ is defined as the ratio of the y-coordinate to the x-coordinate, or $\frac{\sin \theta}{\cos \theta}$. Geometrically, if you draw a tangent line to the circle at the point $(1, 0)$, the value of $\tan \theta$ corresponds to the y-intercept of the line formed by the terminal side of the angle intersecting this tangent line.