12.2 The Unit Circle, Radian Measure & Exact Trigonometric Values

12.2 The Unit Circle, Radian Measure & Exact Trigonometric Values

Radian Measure: Geometric Definition and Angular Dynamics

While degree measure partitions a circular rotation into 360 arbitrary segments rooted in ancient sexagesimal astronomy, the radian provides an intrinsic, mathematically natural measure of angle. The radian measure of a central angle $\theta$ is defined as the ratio of the subtended arc length $s$ to the radius $r$ of the circle:

$$\theta = \frac{s}{r}$$

Because arc length and radius share identical metric units of distance (e.g., meters or inches), radian measure is strictly dimensionless. An angle of 1 radian subtends an arc whose circular boundary length exactly equals the radius of the circle.

From the circumference formula of a circle, $C = 2\pi r$, a complete revolution of $360^\circ$ corresponds to an arc length of $2\pi r$, yielding:

$$\theta_{\text{full}} = \frac{2\pi r}{r} = 2\pi\text{ radians}$$

This identity establishes the fundamental conversion equivalence:

$$180^\circ = \pi\text{ radians}$$

To convert degrees to radians, multiply the degree measure by $\frac{\pi}{180^\circ}$:

$$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180^\circ}$$

To convert radians to degrees, multiply the radian measure by $\frac{180^\circ}{\pi}$:

$$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi}$$

Radian measure simplifies two vital metric formulas:

1. Arc Length: $s = r\theta$, where $\theta$ is expressed in radians.
2. Sector Area: Slicing a circular sector of angle $\theta$ from total circle area $\pi r^2$ gives the proportional relation $\frac{A}{\pi r^2} = \frac{\theta}{2\pi}$, yielding: $$A = \frac{1}{2}r^2\theta$$

In dynamic systems, an object rotating with angular velocity $\omega = \frac{d\theta}{dt}$ (in radians per unit time) traces an arc with linear tangential velocity:

$$v = \frac{ds}{dt} = r\frac{d\theta}{dt} = r\omega$$

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The Unit Circle and Circular Trigonometric Definitions

The unit circle is the circle of radius $r = 1$ centered at the origin of the Cartesian coordinate plane, governed by the algebraic equation:

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

An angle $\theta$ is in standard position when its vertex resides at the origin $(0, 0)$ and its initial ray coincides with the positive x-axis. As the terminal ray sweeps counterclockwise (for positive angles) or clockwise (for negative angles), it intersects the unit circle at a unique terminal point $P(x, y)$.

By constructing a reference triangle with hypotenuse $r = 1$, horizontal adjacent leg $x$, and vertical opposite leg $y$, the circular definitions of the trigonometric functions emerge directly:

$$\cos\theta = x, \quad \sin\theta = y$$

Thus, the terminal coordinates of any real angle $\theta$ on the unit circle are identically:

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

From these coordinates, the remaining four trigonometric functions are defined as algebraic ratios:

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

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The Fundamental Pythagorean Identities

Substituting $x = \cos\theta$ and $y = \sin\theta$ directly into the Cartesian unit circle equation $x^2 + y^2 = 1$ establishes the primary Pythagorean trigonometric identity:

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

Dividing this fundamental identity by $\cos^2\theta$ (under the domain restriction $\cos\theta \neq 0$) generates the second Pythagorean identity:

$$\frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta} \implies \tan^2\theta + 1 = \sec^2\theta$$

Dividing the fundamental identity by $\sin^2\theta$ (under the restriction $\sin\theta \neq 0$) produces the third Pythagorean identity:

$$\frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} \implies 1 + \cot^2\theta = \csc^2\theta$$

These three identities allow instantaneous algebraic conversions between trigonometric expressions when analyzing expressions, proving identities, or evaluating integrals in calculus.

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Reference Angles and Quadrantal Signs (ASTC)

A reference angle $\theta'$ is the positive acute angle formed between the terminal ray of angle $\theta$ and the horizontal x-axis ($0 \le \theta' \le \frac{\pi}{2}$ or $0^\circ \le \theta' \le 90^\circ$). Because reference triangles across all four quadrants are congruent up to reflection across the coordinate axes, any trigonometric value of $\theta$ matches the value of its reference angle $\theta'$ in absolute magnitude, differing only by an algebraic sign determined by the quadrant.

Reference angle reduction formulas for angles $0 \le \theta < 2\pi$:

• Quadrant I: $\theta' = \theta$
• Quadrant II: $\theta' = \pi - \theta = 180^\circ - \theta$
• Quadrant III: $\theta' = \theta - \pi = \theta - 180^\circ$
• Quadrant IV: $\theta' = 2\pi - \theta = 360^\circ - \theta$

The algebraic signs of trigonometric functions across the four quadrants are codified by the traditional mnemonic ASTC ("All Students Take Calculus"):

1. Quadrant I ($0 < \theta < \frac{\pi}{2}$): Both $x > 0$ and $y > 0$. All six trigonometric functions are positive.
2. Quadrant II ($\frac{\pi}{2} < \theta < \pi$): Here $x < 0$ and $y > 0$. Sine and cosecant are positive; cosine, secant, tangent, and cotangent are negative.
3. Quadrant III ($\pi < \theta < \frac{3\pi}{2}$): Here $x < 0$ and $y < 0$. Tangent and cotangent are positive (as $\frac{-y}{-x} > 0$); sine, cosecant, cosine, and secant are negative.
4. Quadrant IV ($\frac{3\pi}{2} < \theta < 2\pi$): Here $x > 0$ and $y < 0$. Cosine and secant are positive; sine, cosecant, tangent, and cotangent are negative.

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Worked Exemplar: Multi-Quadrant Exact Trigonometric Evaluation

Problem: Evaluate the exact value of $\tan\left(\frac{19\pi}{6}\right) - \sec\left(-\frac{7\pi}{4}\right) + \csc\left(\frac{5\pi}{2}\right)$.

• Step 1: Simplify Angle 1: $\frac{19\pi}{6}$. Subtract complete revolutions of $2\pi = \frac{12\pi}{6}$ to obtain a coterminal angle within $[0, 2\pi)$:

  $$\frac{19\pi}{6} - \frac{12\pi}{6} = \frac{7\pi}{6}$$

  The angle $\frac{7\pi}{6}$ lies in Quadrant III ($\pi < \frac{7\pi}{6} < \frac{3\pi}{2}$). The reference angle is:

  $$\theta' = \frac{7\pi}{6} - \pi = \frac{\pi}{6}$$

  In Quadrant III, tangent is positive (ASTC). Therefore:

  $$\tan\left(\frac{19\pi}{6}\right) = \tan\left(\frac{7\pi}{6}\right) = +\tan\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}$$

• Step 2: Simplify Angle 2: $-\frac{7\pi}{4}$. Add a full revolution of $2\pi = \frac{8\pi}{4}$ to find the standard coterminal angle:

  $$-\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4}$$

  The angle $\frac{\pi}{4}$ lies in Quadrant I. Recall that cosine is an even function: $\cos(-\theta) = \cos\theta \implies \sec(-\theta) = \sec\theta$.

  $$\sec\left(-\frac{7\pi}{4}\right) = \sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos(\pi/4)} = \frac{1}{\sqrt{2}/2} = \sqrt{2}$$

• Step 3: Simplify Angle 3: $\frac{5\pi}{2}$. Subtract $2\pi = \frac{4\pi}{2}$:

  $$\frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}$$

  This is a quadrantal angle terminating on the positive y-axis at point $(0, 1)$. Thus, $\sin\left(\frac{\pi}{2}\right) = 1$, giving:

  $$\csc\left(\frac{5\pi}{2}\right) = \frac{1}{\sin(\pi/2)} = \frac{1}{1} = 1$$

• Step 4: Combine the Exact Radical Terms. Substitute the three exact evaluations into the original expression:

  $$\tan\left(\frac{19\pi}{6}\right) - \sec\left(-\frac{7\pi}{4}\right) + \csc\left(\frac{5\pi}{2}\right) = \frac{\sqrt{3}}{3} - \sqrt{2} + 1 = \frac{\sqrt{3} - 3\sqrt{2} + 3}{3}$$

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Complete Unit Circle Coordinate and Trigonometric Reference Table