Graphs of Trigonometric Functions

You have seen sine and cosine defined on the unit circle. You know that as an angle sweeps around the circle, these functions trace out values between -1 and 1. But what happens when you take those values and plot them on a coordinate plane? Something beautiful emerges: smooth, repeating waves that describe everything from ocean tides to sound vibrations to the position of a swinging pendulum.

If you have ever watched waves roll onto a beach or noticed how daylight hours rise and fall throughout the year, you have witnessed trigonometric graphs in action. These functions capture the essence of any phenomenon that cycles, repeats, or oscillates. In this chapter, we are going to explore what these graphs look like, why they behave the way they do, and how to transform them to model real-world situations.

Core Concepts

The Graph of y = sin(x)

Let us start with the most fundamental trigonometric graph. Remember that $\sin(x)$ gives you the y-coordinate of a point on the unit circle corresponding to angle $x$. As you trace around the unit circle, the height rises, falls, and repeats in a predictable pattern.

Here is what happens as $x$ increases from 0 to $2\pi$:

• At $x = 0$, you are at the point (1, 0) on the unit circle, so $\sin(0) = 0$
• At $x = \frac{\pi}{2}$, you are at (0, 1), so $\sin\left(\frac{\pi}{2}\right) = 1$
• At $x = \pi$, you are at (-1, 0), so $\sin(\pi) = 0$
• At $x = \frac{3\pi}{2}$, you are at (0, -1), so $\sin\left(\frac{3\pi}{2}\right) = -1$
• At $x = 2\pi$, you are back at (1, 0), so $\sin(2\pi) = 0$

When you plot these points and connect them smoothly, you get the characteristic “wave” shape. The graph starts at the origin, rises to a maximum of 1, falls back through zero to a minimum of -1, then rises back to zero to complete one cycle.

Key features of y = sin(x):

• Period: $2\pi$ (the graph repeats every $2\pi$ units)
• Amplitude: 1 (the distance from the midline to a maximum or minimum)
• Midline: $y = 0$ (the horizontal line halfway between maximum and minimum)
• Maximum value: 1
• Minimum value: -1
• Key points in one period: $(0, 0)$, $\left(\frac{\pi}{2}, 1\right)$, $(\pi, 0)$, $\left(\frac{3\pi}{2}, -1\right)$, $(2\pi, 0)$

The Graph of y = cos(x)

The cosine function has the same wave shape as sine, but it starts at a different position. While sine starts at zero and rises, cosine starts at its maximum value and falls.

This makes sense if you think about the unit circle: $\cos(x)$ gives you the x-coordinate of a point. At $x = 0$, you are at (1, 0), so $\cos(0) = 1$.

Key features of y = cos(x):

• Period: $2\pi$
• Amplitude: 1
• Midline: $y = 0$
• Maximum value: 1
• Minimum value: -1
• Key points in one period: $(0, 1)$, $\left(\frac{\pi}{2}, 0\right)$, $(\pi, -1)$, $\left(\frac{3\pi}{2}, 0\right)$, $(2\pi, 1)$

Important relationship: The graphs of sine and cosine are identical in shape; cosine is just sine shifted $\frac{\pi}{2}$ units to the left. In other words: $$\cos(x) = \sin\left(x + \frac{\pi}{2}\right)$$

Transformations: The General Form

The real power of trigonometric functions comes when we transform them. The general form for a transformed sine or cosine function is:

$$y = A \cdot \sin(Bx - C) + D \quad \text{or} \quad y = A \cdot \cos(Bx - C) + D$$

Each parameter controls a different aspect of the graph:

A (Amplitude): This value stretches or compresses the graph vertically.

• $|A|$ is the amplitude (always positive)
• If $A < 0$, the graph is reflected over the x-axis (flipped upside down)
• The graph oscillates between $D - |A|$ and $D + |A|$

B (Frequency factor): This affects how quickly the function cycles.

• Period $= \frac{2\pi}{|B|}$
• Larger $|B|$ means shorter period (more cycles in the same horizontal distance)
• Smaller $|B|$ means longer period (stretched horizontally)

C (Phase shift factor): This shifts the graph horizontally.

• Phase shift $= \frac{C}{B}$
• If $\frac{C}{B} > 0$, the graph shifts right
• If $\frac{C}{B} < 0$, the graph shifts left

D (Vertical shift): This moves the entire graph up or down.

• The midline becomes $y = D$
• Maximum value is $D + |A|$
• Minimum value is $D - |A|$

The Graph of y = tan(x)

Tangent behaves quite differently from sine and cosine. Recall that $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Whenever cosine equals zero, tangent is undefined, which creates vertical asymptotes in the graph.

Key features of y = tan(x):

• Period: $\pi$ (not $2\pi$ like sine and cosine)
• Amplitude: undefined (tangent has no maximum or minimum; it extends to infinity)
• Asymptotes: Vertical lines at $x = \frac{\pi}{2} + n\pi$ for any integer $n$
• Passes through origin: $\tan(0) = 0$
• Shape: Between each pair of asymptotes, the graph rises from $-\infty$ to $+\infty$

The tangent graph looks like a series of increasing curves, each separated by vertical asymptotes. It passes through the origin and increases without bound as it approaches $\frac{\pi}{2}$, while decreasing without bound as it approaches $-\frac{\pi}{2}$.

The Graph of y = cot(x)

Cotangent is the reciprocal of tangent: $\cot(x) = \frac{\cos(x)}{\sin(x)}$. Its asymptotes occur where sine equals zero.

Key features of y = cot(x):

• Period: $\pi$
• Amplitude: undefined
• Asymptotes: Vertical lines at $x = n\pi$ for any integer $n$
• Shape: Between each pair of asymptotes, the graph falls from $+\infty$ to $-\infty$

Cotangent is like a reflected version of tangent, shifted horizontally by $\frac{\pi}{2}$.

The Graphs of y = sec(x) and y = csc(x)

Secant and cosecant are the reciprocals of cosine and sine, respectively: $$\sec(x) = \frac{1}{\cos(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}$$

Because they are reciprocals, their graphs have a special relationship with the original functions.

Key features of y = sec(x):

• Period: $2\pi$
• Asymptotes: Where $\cos(x) = 0$, namely at $x = \frac{\pi}{2} + n\pi$
• Shape: U-shaped curves opening upward and downward, alternating
• Range: $(-\infty, -1] \cup [1, \infty)$ (never between -1 and 1)
• Relationship to cosine: The maximum points of cosine become minimum points of secant, and vice versa

Key features of y = csc(x):

• Period: $2\pi$
• Asymptotes: Where $\sin(x) = 0$, namely at $x = n\pi$
• Shape: U-shaped curves opening upward and downward, alternating
• Range: $(-\infty, -1] \cup [1, \infty)$
• Relationship to sine: Similar to secant’s relationship with cosine

To sketch secant or cosecant, it often helps to first lightly sketch the corresponding cosine or sine graph, then draw the reciprocal curves that touch these graphs at their maximum and minimum points and curve away toward the asymptotes.

Notation and Terminology

Term	Definition
Amplitude	The distance from the midline to a maximum (or minimum). For $y = A\sin(x)$, the amplitude is $|A|$.
Period	The horizontal length of one complete cycle. For $y = \sin(Bx)$, the period is $\frac{2\pi}{|B|}$.
Midline	The horizontal line exactly halfway between the maximum and minimum values. For $y = \sin(x) + D$, the midline is $y = D$.
Phase shift	The horizontal displacement of the graph. For $y = \sin(Bx - C)$, the phase shift is $\frac{C}{B}$.
Vertical shift	The amount the graph moves up or down from the standard position. Given by $D$ in the general form.
Asymptote	A line that the graph approaches but never touches. Tangent, cotangent, secant, and cosecant all have vertical asymptotes.
Frequency	How many complete cycles occur in a given interval. Frequency $= \frac{1}{\text{period}}$.

Examples

Key Properties

Sine and Cosine Properties

Property	$y = \sin(x)$	$y = \cos(x)$
Domain	All real numbers	All real numbers
Range	$[-1, 1]$	$[-1, 1]$
Period	$2\pi$	$2\pi$
Amplitude	1	1
x-intercepts	$x = n\pi$	$x = \frac{\pi}{2} + n\pi$
Maximum	1 at $x = \frac{\pi}{2} + 2n\pi$	1 at $x = 2n\pi$
Minimum	-1 at $x = \frac{3\pi}{2} + 2n\pi$	-1 at $x = \pi + 2n\pi$
Symmetry	Odd function: $\sin(-x) = -\sin(x)$	Even function: $\cos(-x) = \cos(x)$

Tangent and Cotangent Properties

Property	$y = \tan(x)$	$y = \cot(x)$
Domain	All reals except $x = \frac{\pi}{2} + n\pi$	All reals except $x = n\pi$
Range	All real numbers	All real numbers
Period	$\pi$	$\pi$
Asymptotes	$x = \frac{\pi}{2} + n\pi$	$x = n\pi$
x-intercepts	$x = n\pi$	$x = \frac{\pi}{2} + n\pi$
Symmetry	Odd function	Odd function

Secant and Cosecant Properties

Property	$y = \sec(x)$	$y = \csc(x)$
Domain	All reals except $x = \frac{\pi}{2} + n\pi$	All reals except $x = n\pi$
Range	$(-\infty, -1] \cup [1, \infty)$	$(-\infty, -1] \cup [1, \infty)$
Period	$2\pi$	$2\pi$
Asymptotes	$x = \frac{\pi}{2} + n\pi$	$x = n\pi$
Symmetry	Even function	Odd function

Transformation Quick Reference

For $y = A\sin(Bx - C) + D$ or $y = A\cos(Bx - C) + D$:

Parameter	Effect	Formula
$A$	Vertical stretch/compression and reflection	Amplitude = $|A|$
$B$	Horizontal stretch/compression	Period = $\frac{2\pi}{|B|}$
$C$	Horizontal shift	Phase shift = $\frac{C}{B}$
$D$	Vertical shift	Midline: $y = D$

Real-World Applications

Sound Waves

When you hear a sound, your eardrum vibrates in response to pressure waves in the air. Pure tones, like those from a tuning fork, produce sinusoidal waves. The amplitude of the wave corresponds to volume (loudness), and the frequency corresponds to pitch. A violin playing concert A produces a wave that can be modeled by:

$$y = A\sin(880\pi t)$$

where $t$ is time in seconds. The frequency of 440 Hz (cycles per second) comes from the coefficient of $t$: since the period is $\frac{2\pi}{880\pi} = \frac{1}{440}$ seconds, there are 440 cycles per second.

Tides

Ocean tides follow a roughly sinusoidal pattern as the gravitational pull of the Moon (and to a lesser extent, the Sun) raises and lowers sea levels. If a harbor has high tides at 6 AM and 6 PM with a water depth of 14 feet, and low tides at noon and midnight with a depth of 6 feet, the water depth $d$ in feet can be modeled as:

$$d(t) = 4\cos\left(\frac{\pi}{6}t - \pi\right) + 10$$

where $t$ is hours after midnight.

Ferris Wheels

Your height on a Ferris wheel varies sinusoidally with time. If a Ferris wheel has a diameter of 50 meters, its center is 30 meters above the ground, and it takes 2 minutes for one complete revolution, your height $h$ in meters after $t$ minutes (starting at the bottom) can be modeled as:

$$h(t) = -25\cos(\pi t) + 30$$

The negative cosine is used because you start at the minimum (bottom of the wheel).

Electrical Current

Alternating current (AC) in your home outlets follows a sinusoidal pattern. In North America, the voltage oscillates at 60 Hz with a peak of about 170 volts:

$$V(t) = 170\sin(120\pi t)$$

The “120 volts” you hear about is actually the root-mean-square (RMS) value, which is the peak divided by $\sqrt{2}$.

Daylight Hours

The number of daylight hours throughout the year follows an approximately sinusoidal pattern. At a latitude of 40 degrees North, the function might look like:

$$D(t) = 3\sin\left(\frac{2\pi}{365}(t - 80)\right) + 12$$

where $t$ is the day of the year (January 1 = day 1). This gives about 15 hours of daylight at the summer solstice and 9 hours at the winter solstice.

Self-Test Problems

Summary

• Sine and cosine produce smooth wave patterns with period $2\pi$ and amplitude 1 in their basic forms. Cosine starts at a maximum while sine starts at zero going upward.

• The general form $y = A\sin(Bx - C) + D$ allows complete control over the wave:

  • $|A|$ = amplitude (vertical stretch)
  • Period = $\frac{2\pi}{|B|}$ (horizontal stretch)
  • Phase shift = $\frac{C}{B}$ (horizontal shift)
  • $D$ = vertical shift (midline position)

• Tangent and cotangent have period $\pi$ and vertical asymptotes. Tangent rises from $-\infty$ to $+\infty$ between asymptotes; cotangent falls from $+\infty$ to $-\infty$.

• Secant and cosecant are reciprocals of cosine and sine. They have vertical asymptotes where their reciprocal functions equal zero and form U-shaped curves that never enter the interval $(-1, 1)$.

• Writing equations from graphs requires identifying the amplitude, period, phase shift, and vertical shift, then working backwards to find $A$, $B$, $C$, and $D$.

• Real-world applications abound: sound waves, tides, Ferris wheels, electrical current, and seasonal patterns all follow sinusoidal models.

The key to mastering these graphs is practice. Start with the basic shapes of sine and cosine, understand how each parameter transforms the graph, and you will be able to analyze and create trigonometric models for any periodic phenomenon you encounter.