Unit circle and radians Knowledge-Based Study

Study notes

1. Measuring Angles in Radians

   In geometry, angles are frequently measured in degrees, where one full revolution equals 360 degrees. However, in calculus and advanced trigonometry, the radian is the standard unit of angular measure because it naturally relates an angle to the geometry of a circle.

   By definition, an angle of 1 radian at the center of a circle subtends an arc whose length is exactly equal to the radius $r$ of the circle. The relationship between arc length $s$, radius $r$, and central angle $\theta$ (in radians) is given by:

   $$s = r\theta$$

   Because the circumference of a circle with radius $r$ is $C = 2\pi r$, a complete turn of 360 degrees corresponds to an angle of $2\pi$ radians:

   $$360^\circ = 2\pi\text{ rad} \implies 180^\circ = \pi\text{ rad}$$

   Converting Between Degrees and Radians

   • To convert degrees to radians, multiply by $\frac{\pi}{180^\circ}$:
     $$\theta\text{ (rad)} = \theta^\circ \times \frac{\pi}{180^\circ}$$
   • To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$:
     $$\theta^\circ = \theta\text{ (rad)} \times \frac{180^\circ}{\pi}$$

   For instance, converting $60^\circ$ to radians yields $60 \times \frac{\pi}{180} = \frac{\pi}{3}\text{ rad}$.

   Definition of the Unit Circle

   The unit circle is defined as a circle centered at the origin $(0, 0)$ in the Cartesian coordinate plane with a radius of $r = 1$. Its algebraic equation is:

   $$x^2 + y^2 = 1$$

   When an angle $\theta$ is placed in standard position, its vertex sits at $(0, 0)$, its initial side lies along the positive $x$-axis, and its terminal side intersects the unit circle at a point $P(x, y)$.

   Coordinates as Trigonometric Values

   Because the radius is $r = 1$, the geometric definitions of sine and cosine simplify directly to the coordinates of the intersection point $P$:

   $$\cos(\theta) = \frac{x}{r} = x$$$$\sin(\theta) = \frac{y}{r} = y$$

   Consequently, any point on the unit circle can be parameterized by the angle $\theta$:

   $$P(x, y) = (\cos(\theta), \sin(\theta))$$

   From these fundamental coordinates, the remaining trigonometric functions are defined for $\theta$ wherever their denominators are non-zero:

   • $\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\quad (x \neq 0)$
   • $\sec(\theta) = \frac{1}{x} = \frac{1}{\cos(\theta)}\quad (x \neq 0)$
   • $\csc(\theta) = \frac{1}{y} = \frac{1}{\sin(\theta)}\quad (y \neq 0)$
   • $\cot(\theta) = \frac{x}{y} = \frac{\cos(\theta)}{\sin(\theta)}\quad (y \neq 0)$

   Trigonometric Signs Across the Four Quadrants

   As the terminal side rotates counterclockwise through the Cartesian plane, the signs of $x = \cos(\theta)$ and $y = \sin(\theta)$ change according to the quadrant:

   • Quadrant I ($0 < \theta < \frac{\pi}{2}$): $x > 0$, $y > 0$ — All trigonometric ratios are positive.
   • Quadrant II ($\frac{\pi}{2} < \theta < \pi$): $x < 0$, $y > 0$ — Sine and cosecant are positive; cosine and tangent are negative.
   • Quadrant III ($\pi < \theta < \frac{3\pi}{2}$): $x < 0$, $y < 0$ — Tangent and cotangent are positive; sine and cosine are negative.
   • Quadrant IV ($\frac{3\pi}{2} < \theta < 2\pi$): $x > 0$, $y < 0$ — Cosine and secant are positive; sine and tangent are negative.

   Reference Angles

   A reference angle $\theta'$ is the acute angle formed between the terminal side of $\theta$ and the horizontal $x$-axis. Because of circle symmetry, the absolute values of the coordinates of $P(\theta)$ match the coordinates of $P(\theta')$:

   • In Quadrant II: $\theta' = \pi - \theta$
   • In Quadrant III: $\theta' = \theta - \pi$
   • In Quadrant IV: $\theta' = 2\pi - \theta$

   To evaluate any trigonometric ratio for an angle outside Quadrant I, find the ratio for its acute reference angle $\theta'$, then attach the sign determined by the quadrant of the original angle.

   Key Angles on the Unit Circle

   Using special right triangles ($45^\circ\text{--}45^\circ\text{--}90^\circ$ and $30^\circ\text{--}60^\circ\text{--}90^\circ$), the exact coordinates for standard angles in Quadrant I are:

   • $\theta = 0$: $(1, 0)$
   • $\theta = \frac{\pi}{6}$: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
   • $\theta = \frac{\pi}{4}$: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
   • $\theta = \frac{\pi}{3}$: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
   • $\theta = \frac{\pi}{2}$: $(0, 1)$

   Worked Examples

   Example 1: Evaluating an Angle in Quadrant II

   Problem: Determine the exact values of $\cos\left(\frac{5\pi}{6}\right)$ and $\sin\left(\frac{5\pi}{6}\right)$.

   Solution:

   1. Identify the quadrant: $\frac{\pi}{2} < \frac{5\pi}{6} < \pi$, so $\frac{5\pi}{6}$ lies in Quadrant II.
   2. Calculate the reference angle: $\theta' = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$.
   3. Apply standard values: $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ and $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.
   4. Apply quadrant signs: In Quadrant II, cosine is negative and sine is positive.

   Therefore, $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ and $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$.

   Example 2: Negative Angles and Tangent

   Problem: Find the exact value of $\tan\left(-\frac{\pi}{3}\right)$.

   Solution:

   A rotation of $-\frac{\pi}{3}$ places the terminal side in Quadrant IV, co-terminal with $2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$. At this position:

   $$x = \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}, \quad y = \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$$$$\tan\left(-\frac{\pi}{3}\right) = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}$$

   The Fundamental Pythagorean Identity

   Substituting $x = \cos(\theta)$ and $y = \sin(\theta)$ into the equation of the unit circle $x^2 + y^2 = 1$ immediately yields the primary Pythagorean trigonometric identity:

   $$\cos^2(\theta) + \sin^2(\theta) = 1$$

   This identity holds true for every real value of $\theta$.