Chapter 5: Trigonometric Functions

Trigonometric functions are among the most important tools in all of mathematics. They arise naturally from the study of triangles and circles, yet their applications extend far beyond geometry into physics, engineering, music, and data science. In this chapter, we build trigonometry from the ground up: starting with angles and the unit circle, defining all six trigonometric functions, learning to graph them and their transformations, and finally introducing inverse trigonometric functions. If you master this chapter, you will have the foundation needed for calculus and beyond.

5.1 Angles and the Unit Circle

An angle is formed by rotating a ray from an initial side to a terminal side. When we place an angle in standard position, its vertex sits at the origin and its initial side lies along the positive $x$-axis. Angles measured counterclockwise are positive; angles measured clockwise are negative.

Degree and Radian Measure

There are two standard units for measuring angles. Degrees divide one full rotation into 360 equal parts, so a full rotation is $360°$. Radians measure the angle by the length of the arc it subtends on a unit circle. Since the circumference of the unit circle is $2\pi$, a full rotation is $2\pi$ radians.

The conversion between degrees and radians follows from the fact that $360° = 2\pi$ radians:

$$\text{Degrees to radians: } \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$$

$$\text{Radians to degrees: } \theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$$

The Unit Circle

The unit circle is the circle of radius 1 centered at the origin: $x^2 + y^2 = 1$. It is the single most important diagram in trigonometry. Given any angle $\theta$ in standard position, the terminal side intersects the unit circle at a unique point $P$. We define:

$$\cos\theta = x\text{-coordinate of }P, \quad \sin\theta = y\text{-coordinate of }P$$

This definition works for all angles, not just acute angles in right triangles. It unifies trigonometry across all four quadrants and for angles greater than $360°$ or less than $0°$.

Reference Angles

A reference angle is the acute angle $\alpha$ formed between the terminal side of an angle and the $x$-axis. The reference angle always lies between $0°$ and $90°$ (or between $0$ and $\frac{\pi}{2}$ radians). Here is how to find it in each quadrant:

• Quadrant I: $\alpha = \theta$
• Quadrant II: $\alpha = 180° - \theta$ (or $\pi - \theta$)
• Quadrant III: $\alpha = \theta - 180°$ (or $\theta - \pi$)
• Quadrant IV: $\alpha = 360° - \theta$ (or $2\pi - \theta$)

The value of any trigonometric function at angle $\theta$ equals $\pm$ the value of that function at the reference angle $\alpha$. The sign depends on which quadrant $\theta$ is in (we cover the sign rules in Section 5.2).

Special Angles

There are three special angles whose trigonometric values you must memorize. These come from the $30$-$60$-$90$ triangle (half of an equilateral triangle) and the $45$-$45$-$90$ triangle (half of a square cut along the diagonal).

5.2 The Six Trigonometric Functions

Using the unit circle, we have already defined sine and cosine. The remaining four trigonometric functions are built from these two.

Notice that tangent and secant are undefined when $\cos\theta = 0$ (at $\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$), while cosecant and cotangent are undefined when $\sin\theta = 0$ (at $\theta = 0, \pi, 2\pi, \ldots$).

Signs in Each Quadrant: The ASTC Rule

Since $\cos\theta$ gives the $x$-coordinate and $\sin\theta$ gives the $y$-coordinate on the unit circle, the signs of the trigonometric functions depend on the quadrant:

A common mnemonic is "All Students Take Calculus" (ASTC), reading counterclockwise from Quadrant I. In Quadrant I, All functions are positive. In II, only Sine. In III, only Tangent. In IV, only Cosine.

Pythagorean Identities

Since any point $(x, y)$ on the unit circle satisfies $x^2 + y^2 = 1$, we immediately get the most fundamental identity in trigonometry:

5.3 Graphs of Sine and Cosine

Graphing trigonometric functions is one of the most important skills in precalculus. The graphs of sine and cosine are smooth, wave-like curves that repeat every $2\pi$ units. Understanding how to read and transform these graphs is essential for modeling periodic phenomena such as sound waves, tides, and seasonal temperatures.

The Basic Graphs

The graph of $y = \sin x$ starts at the origin, rises to a maximum of $1$ at $x = \frac{\pi}{2}$, returns to $0$ at $x = \pi$, falls to a minimum of $-1$ at $x = \frac{3\pi}{2}$, and returns to $0$ at $x = 2\pi$. The graph of $y = \cos x$ has the same shape but is shifted $\frac{\pi}{2}$ units to the left: it starts at $1$, falls to $0$ at $\frac{\pi}{2}$, reaches $-1$ at $\pi$, returns to $0$ at $\frac{3\pi}{2}$, and returns to $1$ at $2\pi$.

Key properties of the basic graphs:

• Domain: $(-\infty, \infty)$ for both
• Range: $[-1, 1]$ for both
• Period: $2\pi$ (the length of one complete cycle)
• Amplitude: $1$ (the distance from the midline to a peak or trough)

The General Sinusoidal Form

The general form of a sinusoidal function is:

$$y = A\sin\bigl(B(x - C)\bigr) + D$$

Each parameter controls a specific transformation:

Interactive Graph: Exploring Sinusoidal Transformations

Use the sliders below to change the values of $A$, $B$, $C$, and $D$ and watch how the graph of $y = A\sin(B(x - C)) + D$ (blue) transforms compared to the base graph $y = \sin x$ (dashed gray).

Drag the sliders to adjust amplitude, period, phase shift, and vertical shift.

5.4 Graphs of Other Trigonometric Functions

The four remaining trigonometric functions — tangent, cotangent, secant, and cosecant — have graphs that look quite different from the smooth waves of sine and cosine. Their most distinctive feature is the presence of vertical asymptotes where the function is undefined.

The Tangent Function

Since $\tan x = \frac{\sin x}{\cos x}$, the tangent function is undefined wherever $\cos x = 0$, which occurs at $x = \frac{\pi}{2} + n\pi$ for any integer $n$. These produce vertical asymptotes.

Key properties of $y = \tan x$:

• Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$
• Range: $(-\infty, \infty)$
• Period: $\pi$ (not $2\pi$)
• No amplitude (the function is unbounded)
• The graph passes through the origin and is increasing on each branch
• It has an inflection point at each $x$-intercept

Interactive Graph: Tangent Function

The tangent function with its vertical asymptotes shown as dashed lines.

The Cotangent Function

Since $\cot x = \frac{\cos x}{\sin x}$, it is undefined wherever $\sin x = 0$, producing vertical asymptotes at $x = n\pi$. The cotangent also has period $\pi$ but decreases on each branch, which is the opposite behavior of tangent.

Secant and Cosecant

The secant function $y = \sec x = \frac{1}{\cos x}$ has vertical asymptotes wherever $\cos x = 0$ and its graph consists of U-shaped and inverted-U-shaped branches opening away from the $x$-axis. Similarly, $y = \csc x = \frac{1}{\sin x}$ has vertical asymptotes wherever $\sin x = 0$. Both secant and cosecant have period $2\pi$ and range $(-\infty, -1] \cup [1, \infty)$.

5.5 Inverse Trigonometric Functions

Since trigonometric functions are periodic, they are not one-to-one on their entire domains and therefore do not have inverses unless we restrict their domains. By choosing a specific interval on which the function is one-to-one, we can define inverse trigonometric functions that "undo" the original function.

The restricted domains are chosen carefully so that each inverse function gives a unique output for every input. For arcsine and arctangent, the range is centered around $0$. For arccosine, the range starts at $0$ and goes to $\pi$, which covers the top half of the unit circle.

Evaluating Inverse Trig Functions (Exact Values)

To evaluate an expression like $\arcsin\left(\frac{\sqrt{3}}{2}\right)$, ask: "What angle $\theta$ in the range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ has $\sin\theta = \frac{\sqrt{3}}{2}$?" The answer is $\frac{\pi}{3}$, since $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$ and $\frac{\pi}{3}$ is in the required range.

For $\arccos\left(-\frac{1}{2}\right)$, we need an angle in $[0, \pi]$ with cosine equal to $-\frac{1}{2}$. The reference angle is $\frac{\pi}{3}$ (since $\cos\frac{\pi}{3} = \frac{1}{2}$), and to get a negative cosine we need Quadrant II: $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.

5.6 Practice Problems

Test your understanding of Chapter 5 with these problems. Each has a full solution that you can reveal by clicking "Show Solution."