7.2 Unit Circle and Trigonometric Function Graphs
Key Takeaways
- On the unit circle (x^2 + y^2 = 1), any real angle t defines coordinates (x, y) = (\cos t, \sin t) and ratio \tan t = y/x (x \neq 0).
- Reference angles \theta_R map angle values from Quadrants II, III, and IV back to Quadrant I, with signs determined by the ASTC rule.
- Sine and cosine functions are periodic with period 2\pi, while tangent and cotangent have a fundamental period of \pi.
- For transformed sine/cosine functions y = A\sin(Bx - C) + D, the amplitude is |A|, period is \frac{2\pi}{|B|}, phase shift is \frac{C}{B}, and vertical shift is D.
- Vertical asymptotes occur where trigonometric functions are undefined, such as x = \frac{\pi}{2} + k\pi for \tan x and \sec x, and x = k\pi for \cot x and \csc x.
Last updated: August 2026
7.2 Unit Circle and Trigonometric Function Graphs
Extending trigonometric functions beyond acute right-triangle angles requires the concept of the unit circle. By defining trigonometric ratios as coordinates on a circle of radius $1$, trigonometry expands to all real numbers, giving rise to continuous, periodic functions.
1. The Unit Circle and Quadrant Signs
The unit circle is centered at the origin $(0,0)$ in the Cartesian plane with a radius of $r = 1$. Its algebraic equation is:
If a real number $t$ represents the directed arc length along the unit circle from $(1,0)$ (which corresponds to a central angle of $t$ radians), the terminal point $P(x,y)$ on the unit circle defines all six trigonometric functions:
\cos t &= x \quad & \sec t &= \frac{1}{x} \quad (x \neq 0) \\ \sin t &= y \quad & \csc t &= \frac{1}{y} \quad (y \neq 0) \\ \tan t &= \frac{y}{x} \quad (x \neq 0) \quad & \cot t &= \frac{x}{y} \quad (y \neq 0) \end{aligned}$$ ### Quadrant Signs (ASTC Rule) The algebraic sign ($+$ or $-$) of each function depends entirely on the quadrant in which the terminal point $P(x,y)$ lies. The acronym **ASTC** ("All Students Take Calculus") summarizes positive functions: - **Quadrant I** ($0 < t < \frac{\pi}{2}$): **A**ll functions are positive ($x > 0, y > 0$). - **Quadrant II** ($\frac{\pi}{2} < t < \pi$): **S**ine and cosecant are positive ($x < 0, y > 0$). - **Quadrant III** ($\pi < t < \frac{3\pi}{2}$): **T**angent and cotangent are positive ($x < 0, y < 0$). - **Quadrant IV** ($\frac{3\pi}{2} < t < 2\pi$): **C**osine and secant are positive ($x > 0, y < 0$). ### Reference Angles For any non-quadrantal angle $\theta$, its **reference angle** $\theta_R$ (or $\theta'$) is the acute angle formed between the terminal side of $\theta$ and the **x-axis**: - **Quadrant II**: $\theta_R = \pi - \theta \quad (180^\circ - \theta)$ - **Quadrant III**: $\theta_R = \theta - \pi \quad (\theta - 180^\circ)$ - **Quadrant IV**: $\theta_R = 2\pi - \theta \quad (360^\circ - \theta)$ To evaluate $\text{Trig}(\theta)$, determine the sign using the ASTC rule for $\theta$'s quadrant and evaluate $\text{Trig}(\theta_R)$. #### Worked Example 1: Evaluating Unit Circle Angles Evaluate the exact value of $\sin\left(\frac{7\pi}{6}\right) + \cos\left(\frac{4\pi}{3}\right)$. *Solution*: 1. For $\frac{7\pi}{6}$: - Lies in Quadrant III ($\pi < \frac{7\pi}{6} < \frac{3\pi}{2}$). Sine is negative in QIII. - Reference angle: $\theta_R = \frac{7\pi}{6} - \pi = \frac{\pi}{6}$. - Value: $\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$. 2. For $\frac{4\pi}{3}$: - Lies in Quadrant III ($\pi < \frac{4\pi}{3} < \frac{3\pi}{2}$). Cosine is negative in QIII. - Reference angle: $\theta_R = \frac{4\pi}{3} - \pi = \frac{\pi}{3}$. - Value: $\cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}$. 3. Combine results: $$\sin\left(\frac{7\pi}{6}\right) + \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} + \left(-\frac{1}{2}\right) = -1$$ --- ## 2. Domain, Range, Periodicity, and Symmetry The structural properties of the trigonometric functions are summarized below: | Function | Domain | Range | Fundamental Period | Symmetry (Even/Odd) | | :--- | :--- | :--- | :--- | :--- | | $y = \sin x$ | $(-\infty, \infty)$ | $[-1, 1]$ | $2\pi$ | Odd: $\sin(-x) = -\sin x$ | | $y = \cos x$ | $(-\infty, \infty)$ | $[-1, 1]$ | $2\pi$ | Even: $\cos(-x) = \cos x$ | | $y = \tan x$ | $x \neq \frac{\pi}{2} + k\pi$ | $(-\infty, \infty)$ | $\pi$ | Odd: $\tan(-x) = -\tan x$ | | $y = \csc x$ | $x \neq k\pi$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ | Odd: $\csc(-x) = -\csc x$ | | $y = \sec x$ | $x \neq \frac{\pi}{2} + k\pi$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ | Even: $\sec(-x) = \sec x$ | | $y = \cot x$ | $x \neq k\pi$ | $(-\infty, \infty)$ | $\pi$ | Odd: $\cot(-x) = -\cot x$ | --- ## 3. Transformations of Sine and Cosine Graphs General sinusoidal functions are modeled by: $$y = A \sin(Bx - C) + D \quad \text{or} \quad y = A \cos(Bx - C) + D$$ To analyze graph parameters, rewrite in factored standard form: $y = A \sin\left[B\left(x - \frac{C}{B}\right)\right] + D$. - **Amplitude**: $|A|$ — Half the vertical distance between maximum and minimum values. - **Period**: $P = \frac{2\pi}{|B|}$ — Length of one complete cycle along the x-axis. - **Phase Shift**: $\frac{C}{B}$ — Horizontal shift (right if $\frac{C}{B} > 0$, left if $\frac{C}{B} < 0$). - **Vertical Shift**: $D$ — Shifts the midline to $y = D$. - **Range**: $[D - |A|, D + |A|]$. #### Worked Example 2: Analyzing a Transformed Cosine Function For the function $f(x) = -3\cos(4x - \pi) + 2$: 1. State the amplitude, period, phase shift, and vertical shift. 2. Determine the maximum and minimum values and the range. *Solution*: 1. Factor out $B = 4$: $$f(x) = -3\cos\left[4\left(x - \frac{\pi}{4}\right)\right] + 2$$ - **Amplitude**: $|A| = |-3| = 3$. - **Period**: $P = \frac{2\pi}{B} = \frac{2\pi}{4} = \frac{\pi}{2}$. - **Phase Shift**: $\frac{C}{B} = \frac{\pi}{4}$ units to the right. - **Vertical Shift**: $D = 2$ (midline at $y = 2$). 2. Range calculations: - Maximum value: $D + |A| = 2 + 3 = 5$. - Minimum value: $D - |A| = 2 - 3 = -1$. - **Range**: $[-1, 5]$. --- ## 4. Graphs of Tangent, Secant, Cosecant, and Cotangent Unlike sine and cosine, the remaining four trigonometric functions contain division by variables that periodically equal zero, creating vertical asymptotes. ### Tangent and Cotangent Graphs - **$y = \tan x$**: Period is $\pi$. Vertical asymptotes occur at $x = \frac{\pi}{2} + k\pi$. Has x-intercepts at $x = k\pi$. Increases continuously between adjacent asymptotes. - **$y = \cot x$**: Period is $\pi$. Vertical asymptotes occur at $x = k\pi$. Has x-intercepts at $x = \frac{\pi}{2} + k\pi$. Decreases continuously between adjacent asymptotes. ### Secant and Cosecant Graphs Secant and cosecant graphs are constructed using their reciprocal guide curves ($y = \cos x$ and $y = \sin x$): - Local maxima of cosine/sine become local minima for secant/cosecant. - Zero-crossings of cosine/sine become vertical asymptotes for secant/cosecant. #### Worked Example 3: Finding Vertical Asymptotes Find all vertical asymptotes of $g(x) = \tan(2x)$. *Solution*: 1. The standard tangent function $\tan(\theta)$ has vertical asymptotes where its argument $\theta = \frac{\pi}{2} + k\pi$ for $k \in \mathbb{Z}$. 2. Set the inner argument $2x$ equal to $\frac{\pi}{2} + k\pi$: $$2x = \frac{\pi}{2} + k\pi$$ 3. Divide by $2$: $$x = \frac{\pi}{4} + \frac{k\pi}{2} \quad (k \in \mathbb{Z})$$ --- ## 5. CLEP Exam Traps & Common Errors - **Trap 1: Period Calculation for Tangent and Cotangent** Applying $\frac{2\pi}{B}$ to tangent or cotangent functions instead of $\frac{\pi}{B}$ is a frequent error. Sine, cosine, secant, and cosecant have base period $2\pi$, whereas tangent and cotangent have base period $\pi$. - **Trap 2: Unfactored Phase Shift** Reading the phase shift from $y = A\sin(Bx - C)$ directly as $C$ instead of $\frac{C}{B}$. You must factor out $B$ first! - **Trap 3: Reference Angle to the Y-Axis** Reference angles MUST be measured relative to the horizontal **x-axis**, never the vertical y-axis.Test Your Knowledge
What is the exact value of \sin(7\pi/6) + \cos(4\pi/3)?
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What are the period and phase shift of the trigonometric function y = -3\cos(4x - \pi) + 2?
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What is the set of all vertical asymptotes for the function f(x) = \tan(2x)?
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What is the range of the function g(x) = 5\sin(3x - \pi/2) - 2?
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