Trigonometric Functions and Their Graphs
See the beautiful wave patterns that sine and cosine create
Have you ever watched waves roll onto a beach, one after another, in a steady rhythm? Or noticed how the days get longer in summer and shorter in winter, year after year, following the same pattern? Or perhaps you have seen an audio waveform displayed on a screen, those smooth curves rising and falling as music plays?
All of these phenomena share something profound: they repeat. They cycle. They oscillate. And when mathematicians need to describe anything that repeats, they reach for sine and cosine.
So, if the words “sine” and “cosine” seem scary to you, then know that you are not alone. Many students feel intimidated by trigonometric functions, partly because the names sound exotic and partly because they seem so different from the functions you have seen before. But here is the thing: you need to realize something. You already understand repeating patterns. You have been surrounded by them your entire life. What we are doing in this lesson is simply giving you the mathematical tools to describe those patterns precisely.
In earlier work with trigonometry, you may have defined sine and cosine using triangles: the ratio of sides in a right triangle. That definition is useful, but it limits you to angles between 0 and 90 degrees. Here, we break free from that limitation. We will see sine and cosine as functions of any real number, functions that trace out the most beautiful waves in all of mathematics.
Core Concepts
Trig Functions as Functions of Real Numbers
When you first learned about sine and cosine, they were probably defined using right triangles. The sine of an angle was “opposite over hypotenuse,” and the cosine was “adjacent over hypotenuse.” This works fine for acute angles, but what about an angle of 200 degrees? Or negative angles? Or angles larger than 360 degrees?
The key insight is the unit circle. Imagine a circle with radius 1 centered at the origin. Now imagine a point moving around this circle. As the point travels, it traces out an angle $\theta$ measured from the positive x-axis.
Here is the crucial definition:
- $\cos \theta$ is the x-coordinate of the point on the unit circle
- $\sin \theta$ is the y-coordinate of the point on the unit circle
This definition works for any angle, no matter how large, small, or negative. If you walk around the circle and keep going, you come back to where you started. That is why these functions repeat.
When we write $y = \sin x$ or $y = \cos x$, we are using $x$ to represent an angle (measured in radians), and $y$ is the output of the function. The variable $x$ can be any real number. It does not have to be between 0 and $2\pi$. It can be 100, or -50, or any number you like. The function will still give you a meaningful output.
The Graph of y = sin x
Let us trace through what $\sin x$ does as $x$ goes from 0 to $2\pi$:
- At $x = 0$: The point is at $(1, 0)$ on the unit circle, so $\sin 0 = 0$
- At $x = \frac{\pi}{2}$: The point is at $(0, 1)$, so $\sin \frac{\pi}{2} = 1$
- At $x = \pi$: The point is at $(-1, 0)$, so $\sin \pi = 0$
- At $x = \frac{3\pi}{2}$: The point is at $(0, -1)$, so $\sin \frac{3\pi}{2} = -1$
- At $x = 2\pi$: The point returns to $(1, 0)$, so $\sin 2\pi = 0$
If you plot these points and connect them smoothly, you get the classic sine wave. It starts at the origin, rises to a maximum of 1, falls back through zero, drops to a minimum of -1, and returns to zero. Then it repeats, forever.
Key features of $y = \sin x$:
- Starts at the origin: The graph passes through $(0, 0)$
- Maximum value: 1, occurring at $x = \frac{\pi}{2}, \frac{5\pi}{2}, \frac{9\pi}{2}, …$
- Minimum value: -1, occurring at $x = \frac{3\pi}{2}, \frac{7\pi}{2}, \frac{11\pi}{2}, …$
- Zeros: At $x = 0, \pi, 2\pi, 3\pi, …$ (every multiple of $\pi$)
- Period: $2\pi$ (the pattern repeats every $2\pi$ units)
The Graph of y = cos x
The cosine function follows the same logic, but tracks the x-coordinate instead of the y-coordinate:
- At $x = 0$: $\cos 0 = 1$
- At $x = \frac{\pi}{2}$: $\cos \frac{\pi}{2} = 0$
- At $x = \pi$: $\cos \pi = -1$
- At $x = \frac{3\pi}{2}$: $\cos \frac{3\pi}{2} = 0$
- At $x = 2\pi$: $\cos 2\pi = 1$
The cosine wave looks exactly like the sine wave, but shifted. In fact, $\cos x = \sin(x + \frac{\pi}{2})$. The cosine curve starts at its maximum, drops to zero, falls to its minimum, rises back through zero, and returns to its maximum.
Key features of $y = \cos x$:
- Starts at maximum: The graph passes through $(0, 1)$
- Maximum value: 1, occurring at $x = 0, 2\pi, 4\pi, …$
- Minimum value: -1, occurring at $x = \pi, 3\pi, 5\pi, …$
- Zeros: At $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, …$ (odd multiples of $\frac{\pi}{2}$)
- Period: $2\pi$
Key Features: Amplitude, Period, and Midline
To describe any wave, you need to know three things: how tall is it, how often does it repeat, and where is it centered? These correspond to amplitude, period, and midline.
Amplitude is half the distance from the maximum to the minimum. For $y = \sin x$ and $y = \cos x$, the maximum is 1 and the minimum is -1, so the amplitude is $\frac{1 - (-1)}{2} = 1$.
Think of amplitude as “how far the wave travels from its center.” A wave with amplitude 3 swings from 3 above the midline to 3 below it.
Period is the horizontal length of one complete cycle. For $y = \sin x$ and $y = \cos x$, the period is $2\pi$. This means if you walk $2\pi$ units along the x-axis, the wave pattern repeats exactly.
Period answers the question: “How long until the pattern starts over?”
Midline is the horizontal line exactly halfway between the maximum and minimum values. For $y = \sin x$ and $y = \cos x$, the midline is $y = 0$ (the x-axis).
If you imagine the wave as water sloshing in a tank, the midline is the resting water level when everything settles down.
Transformations: The General Sinusoidal Form
Now comes the powerful part. We can stretch, shrink, shift, and flip these basic waves using the general form:
$$y = A \sin(Bx - C) + D$$
or equivalently:
$$y = A \cos(Bx - C) + D$$
Each parameter controls a specific transformation:
Amplitude: $|A|$
The coefficient $A$ stretches or compresses the wave vertically.
- If $|A| > 1$, the wave is taller (stretched vertically)
- If $|A| < 1$, the wave is shorter (compressed vertically)
- If $A < 0$, the wave is also reflected across the x-axis (flipped upside down)
The amplitude is always $|A|$, the absolute value, because amplitude measures distance, which is always positive.
Period: $\frac{2\pi}{|B|}$
The coefficient $B$ stretches or compresses the wave horizontally.
- If $|B| > 1$, the wave completes more cycles in the same space (compressed horizontally, shorter period)
- If $|B| < 1$, the wave completes fewer cycles (stretched horizontally, longer period)
Why $\frac{2\pi}{|B|}$? Because the basic sine wave has period $2\pi$, and multiplying $x$ by $B$ makes everything happen $B$ times as fast.
Phase Shift: $\frac{C}{B}$
The value $C$ shifts the wave left or right.
- If $\frac{C}{B} > 0$, the wave shifts to the right
- If $\frac{C}{B} < 0$, the wave shifts to the left
Be careful here: the phase shift is $\frac{C}{B}$, not just $C$. This is because the shift happens to $Bx$, not just $x$.
Another way to think about it: the wave “starts” (in the sense of $\sin 0 = 0$) when $Bx - C = 0$, which means $x = \frac{C}{B}$.
Vertical Shift: $D$
The constant $D$ moves the entire wave up or down.
- If $D > 0$, the wave shifts up
- If $D < 0$, the wave shifts down
The midline of the transformed function is $y = D$.
The Graph of y = tan x
The tangent function is the third major trigonometric function, defined as:
$$\tan x = \frac{\sin x}{\cos x}$$
Because tangent is a ratio, and because $\cos x = 0$ at certain points, the tangent function has a fundamentally different character than sine and cosine.
Key features of $y = \tan x$:
Asymptotes: The function is undefined wherever $\cos x = 0$, which happens at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, …$ In general, at $x = \frac{\pi}{2} + n\pi$ for any integer $n$.
Near these x-values, the tangent function shoots off toward infinity (or negative infinity). The graph approaches but never touches these vertical lines, called vertical asymptotes.
Period: Unlike sine and cosine, tangent has period $\pi$, not $2\pi$. The pattern repeats every $\pi$ units.
Zeros: $\tan x = 0$ wherever $\sin x = 0$, which is at $x = 0, \pi, 2\pi, …$ (multiples of $\pi$).
Behavior: Between each pair of asymptotes, the tangent function rises from $-\infty$ to $+\infty$, passing through zero in the middle. It looks like an endless series of S-shaped curves.
No amplitude: Because tangent has no maximum or minimum value (it goes to infinity in both directions), we do not define an amplitude for it.
Domain and Range of Trigonometric Functions
Understanding domain and range is essential for working with these functions.
For $y = \sin x$:
- Domain: All real numbers $(-\infty, \infty)$
- Range: $[-1, 1]$
For $y = \cos x$:
- Domain: All real numbers $(-\infty, \infty)$
- Range: $[-1, 1]$
For $y = \tan x$:
- Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
- Range: All real numbers $(-\infty, \infty)$
When you transform these functions, the domain and range change accordingly:
For $y = A \sin(Bx - C) + D$ or $y = A \cos(Bx - C) + D$:
- Domain: Still all real numbers
- Range: $[D - |A|, D + |A|]$
The vertical shift $D$ moves the center, and the amplitude $|A|$ determines how far above and below that center the wave extends.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Amplitude | Half the distance from max to min | In $y = 3 \sin x$, amplitude is 3 |
| Period | Horizontal length of one complete cycle | Period of $\sin x$ is $2\pi$ |
| Phase shift | Horizontal shift | In $y = \sin(x - \frac{\pi}{2})$, shift is $\frac{\pi}{2}$ right |
| Midline | Horizontal line halfway between max and min | In $y = \sin x + 2$, midline is $y = 2$ |
| Asymptote | Line the graph approaches but never touches | $\tan x$ has asymptotes at $x = \frac{\pi}{2} + n\pi$ |
| Sinusoidal | Having the shape of a sine wave | Both $y = \sin x$ and $y = \cos x$ are sinusoidal |
| Frequency | Number of cycles per unit (reciprocal of period) | If period is $\pi$, frequency is $\frac{1}{\pi}$ |
Examples
Identify the amplitude and period of $y = 2 \sin x$.
Solution:
Compare with the general form $y = A \sin(Bx - C) + D$:
- $A = 2$
- $B = 1$ (since $\sin x = \sin(1 \cdot x)$)
- $C = 0$
- $D = 0$
Amplitude: $|A| = |2| = 2$
Period: $\frac{2\pi}{|B|} = \frac{2\pi}{1} = 2\pi$
Interpretation: This wave oscillates between -2 and 2 (twice as tall as the basic sine wave) and completes one full cycle every $2\pi$ units.
Sketch one complete period of $y = \cos x$.
Solution:
We need to plot key points and connect them smoothly.
Key points for one period (from $x = 0$ to $x = 2\pi$):
| $x$ | $\cos x$ |
|---|---|
| $0$ | $1$ (maximum) |
| $\frac{\pi}{2}$ | $0$ (zero, going down) |
| $\pi$ | $-1$ (minimum) |
| $\frac{3\pi}{2}$ | $0$ (zero, going up) |
| $2\pi$ | $1$ (back to maximum) |
Sketch description:
- Start at the point $(0, 1)$ - this is the maximum
- Draw a smooth curve down to $(\frac{\pi}{2}, 0)$
- Continue down to $(\pi, -1)$ - this is the minimum
- Rise smoothly to $(\frac{3\pi}{2}, 0)$
- Rise to $(2\pi, 1)$ - back to maximum
The curve should be smooth and wavelike, not jagged or with corners. The shape resembles a gentle rolling hill followed by a valley.
Key features visible in the sketch:
- Amplitude: 1 (distance from midline to peak)
- Period: $2\pi$ (distance for one complete cycle)
- Midline: $y = 0$
- Starts at maximum (this distinguishes cosine from sine)
Find the amplitude, period, and phase shift of $y = 3 \sin(2x - \pi)$.
Solution:
First, identify the parameters by comparing with $y = A \sin(Bx - C) + D$:
- $A = 3$
- $B = 2$
- $C = \pi$
- $D = 0$
Amplitude: $$|A| = |3| = 3$$
The wave oscillates from -3 to 3.
Period: $$\frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi$$
The wave completes one full cycle every $\pi$ units (half as long as the basic sine wave).
Phase shift: $$\frac{C}{B} = \frac{\pi}{2}$$
Since this is positive, the wave shifts $\frac{\pi}{2}$ units to the right.
Interpretation: Compared to $y = \sin x$, this wave is 3 times as tall, completes cycles twice as fast, and starts $\frac{\pi}{2}$ units to the right.
Finding where the cycle begins: The standard sine function starts at $(0, 0)$ going up. For $y = 3\sin(2x - \pi)$, this “starting point” occurs when $2x - \pi = 0$, which means $x = \frac{\pi}{2}$. So the wave passes through $(\frac{\pi}{2}, 0)$ on its way up.
Write an equation for a sine wave with amplitude 4 and period $\pi$.
Solution:
We need to find $A$ and $B$ in $y = A \sin(Bx)$.
Find A from the amplitude:
The amplitude is $|A|$. We are told the amplitude is 4, so $|A| = 4$.
We can choose $A = 4$ (unless we need the wave to be reflected, in which case we would use $A = -4$).
Find B from the period:
The period is $\frac{2\pi}{|B|}$. We are told the period is $\pi$, so:
$$\pi = \frac{2\pi}{|B|}$$
Solve for $|B|$: $$|B| = \frac{2\pi}{\pi} = 2$$
We can choose $B = 2$.
Write the equation: $$y = 4 \sin(2x)$$
Verification:
- Amplitude: $|4| = 4$ ✓
- Period: $\frac{2\pi}{2} = \pi$ ✓
Note: This is not the only answer. We could also write $y = -4\sin(2x)$ (reflected) or $y = 4\sin(-2x)$ (which equals $-4\sin(2x)$ by the odd function property). The equation $y = 4\cos(2x - \frac{\pi}{2})$ would also work, since cosine shifted by $\frac{\pi}{2}$ equals sine.
Graph $y = -2\cos\left(x + \frac{\pi}{4}\right) + 1$ showing all transformations.
Solution:
First, rewrite in standard form $y = A\cos(Bx - C) + D$:
$$y = -2\cos\left(x - \left(-\frac{\pi}{4}\right)\right) + 1$$
So: $A = -2$, $B = 1$, $C = -\frac{\pi}{4}$, $D = 1$
Step 1: Identify all transformations
Amplitude: $|A| = |-2| = 2$
Period: $\frac{2\pi}{|B|} = \frac{2\pi}{1} = 2\pi$
Phase shift: $\frac{C}{B} = \frac{-\frac{\pi}{4}}{1} = -\frac{\pi}{4}$ (shift $\frac{\pi}{4}$ to the left)
Vertical shift: $D = 1$ (shift up 1 unit)
Reflection: Since $A = -2 < 0$, the wave is reflected across its midline.
Step 2: Determine key values
Midline: $y = D = 1$
Maximum value: $D + |A| = 1 + 2 = 3$
Minimum value: $D - |A| = 1 - 2 = -1$
Step 3: Find key points
For a standard cosine function, key points in one period are at $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.
But we have a phase shift of $-\frac{\pi}{4}$, so our key points start at $x = -\frac{\pi}{4}$:
| Phase | $x$-value | Before reflection | After reflection ($A < 0$) | Final $y$ (add $D = 1$) |
|---|---|---|---|---|
| Start (max for cos) | $-\frac{\pi}{4}$ | Would be max | Becomes min | $-1$ |
| Quarter | $-\frac{\pi}{4} + \frac{\pi}{2} = \frac{\pi}{4}$ | Would be 0 | Still 0 | $1$ |
| Half | $-\frac{\pi}{4} + \pi = \frac{3\pi}{4}$ | Would be min | Becomes max | $3$ |
| Three-quarter | $-\frac{\pi}{4} + \frac{3\pi}{2} = \frac{5\pi}{4}$ | Would be 0 | Still 0 | $1$ |
| End | $-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}$ | Would be max | Becomes min | $-1$ |
Key points to plot:
- $\left(-\frac{\pi}{4}, -1\right)$ - minimum
- $\left(\frac{\pi}{4}, 1\right)$ - on midline, going up
- $\left(\frac{3\pi}{4}, 3\right)$ - maximum
- $\left(\frac{5\pi}{4}, 1\right)$ - on midline, going down
- $\left(\frac{7\pi}{4}, -1\right)$ - minimum
Step 4: Sketch the graph
Draw a smooth wave through these points. Because of the reflection, this wave starts at its minimum (rather than the usual cosine starting at maximum), rises to the midline, peaks at the maximum, falls back to the midline, and returns to the minimum.
The wave oscillates between $y = -1$ and $y = 3$, centered on the midline $y = 1$.
A sinusoidal wave has:
- Maximum value: 5
- Minimum value: -1
- Period: $4\pi$
- The maximum occurs at $x = \pi$
Write an equation for this wave.
Solution:
Step 1: Find amplitude and midline
The midline is halfway between max and min: $$D = \frac{5 + (-1)}{2} = \frac{4}{2} = 2$$
The amplitude is the distance from midline to max (or min): $$|A| = 5 - 2 = 3$$
So $A = 3$ or $A = -3$ (we will determine which shortly).
Step 2: Find B from the period
$$\text{Period} = \frac{2\pi}{|B|}$$ $$4\pi = \frac{2\pi}{|B|}$$ $$|B| = \frac{2\pi}{4\pi} = \frac{1}{2}$$
So $B = \frac{1}{2}$.
Step 3: Determine the phase shift
We have a choice: use sine or cosine as our base function. Let us use cosine since we know where the maximum is.
For $y = A\cos(Bx - C) + D$, the maximum of the basic $\cos$ function occurs when $Bx - C = 0$, i.e., at $x = \frac{C}{B}$.
We want the maximum at $x = \pi$, so: $$\frac{C}{B} = \pi$$ $$C = B \cdot \pi = \frac{1}{2} \cdot \pi = \frac{\pi}{2}$$
Step 4: Determine the sign of A
Since we want a maximum at $x = \pi$ (not a minimum), and cosine naturally has a maximum when its argument is 0, we use $A = 3$ (positive).
Step 5: Write and verify the equation
$$y = 3\cos\left(\frac{1}{2}x - \frac{\pi}{2}\right) + 2$$
Verification:
Check at $x = \pi$: $$y = 3\cos\left(\frac{1}{2}(\pi) - \frac{\pi}{2}\right) + 2 = 3\cos(0) + 2 = 3(1) + 2 = 5$$ ✓ (maximum)
Check the period: $\frac{2\pi}{|1/2|} = 4\pi$ ✓
Check the range: midline $\pm$ amplitude = $2 \pm 3$, so from $-1$ to $5$ ✓
Alternative form: We could also write this using sine with a different phase shift: $$y = 3\sin\left(\frac{1}{2}x\right) + 2$$
Check: At $x = \pi$, $y = 3\sin(\frac{\pi}{2}) + 2 = 3(1) + 2 = 5$ ✓ (This works because sine reaches its maximum at $\frac{\pi}{2}$, which happens when $x = \pi$.)
Key Properties and Rules
Summary of Transformations
For $y = A\sin(Bx - C) + D$ or $y = A\cos(Bx - C) + D$:
| Parameter | Effect | Formula/Value |
|---|---|---|
| $|A|$ | Amplitude (vertical stretch/compression) | Amplitude $= |A|$ |
| $A < 0$ | Reflection across midline | Wave is flipped |
| $B$ | Horizontal stretch/compression | Period $= \frac{2\pi}{|B|}$ |
| $C$ | Phase shift (horizontal shift) | Shift $= \frac{C}{B}$ to the right |
| $D$ | Vertical shift | Midline is $y = D$ |
Fundamental Values to Remember
| $x$ | $\sin x$ | $\cos x$ | $\tan x$ |
|---|---|---|---|
| $0$ | $0$ | $1$ | $0$ |
| $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ |
| $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $\frac{\pi}{2}$ | $1$ | $0$ | undefined |
| $\pi$ | $0$ | $-1$ | $0$ |
Domain and Range Summary
| Function | Domain | Range |
|---|---|---|
| $y = \sin x$ | All real numbers | $[-1, 1]$ |
| $y = \cos x$ | All real numbers | $[-1, 1]$ |
| $y = \tan x$ | All reals except $\frac{\pi}{2} + n\pi$ | All real numbers |
| $y = A\sin(Bx-C)+D$ | All real numbers | $[D - |A|, D + |A|]$ |
| $y = A\cos(Bx-C)+D$ | All real numbers | $[D - |A|, D + |A|]$ |
Relationships Between Sine and Cosine
These identities are useful for converting between forms:
- $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ (cosine is sine shifted left by $\frac{\pi}{2}$)
- $\sin x = \cos\left(x - \frac{\pi}{2}\right)$ (sine is cosine shifted right by $\frac{\pi}{2}$)
- $\sin(-x) = -\sin x$ (sine is an odd function)
- $\cos(-x) = \cos x$ (cosine is an even function)
Real-World Applications
Sound Waves and Music
Every sound you hear is a pressure wave traveling through the air, and simple sounds (pure tones) are modeled by sine waves. The frequency of the wave (how many cycles per second) determines the pitch you hear. Middle C on a piano corresponds to about 262 cycles per second (262 Hz). Higher frequencies mean higher pitches.
The amplitude of the wave corresponds to loudness. A louder sound has a taller wave; a quieter sound has a shorter wave.
When a guitar string vibrates, it does not produce a single sine wave but a combination of many frequencies. The blend of these frequencies gives each instrument its unique sound (its “timbre”). Understanding this requires Fourier analysis, which breaks complex waves into sums of simple sine and cosine waves.
Ocean Tides
The rise and fall of ocean tides follows a roughly sinusoidal pattern. High tide and low tide alternate approximately every 12 hours and 25 minutes (half a lunar day). The amplitude of the tide varies with the moon’s phase: spring tides (during full and new moons) have larger amplitudes, while neap tides (during quarter moons) have smaller amplitudes.
Coastal cities use tide tables, which are essentially predictions based on sinusoidal models, to help ships navigate harbors and to alert people about flooding risks.
Seasonal Daylight Hours
The number of hours of daylight in a day varies sinusoidally throughout the year. Near the equator, the variation is small. But at higher latitudes, the difference is dramatic: in far northern locations, summer brings nearly 24 hours of daylight, while winter brings almost none.
A typical model for daylight hours is:
$$D(t) = A\sin\left(\frac{2\pi}{365}(t - 80)\right) + B$$
where $t$ is the day of the year, $A$ depends on latitude, and $B$ is the average hours of daylight (about 12).
AC Electrical Current
The electricity in your home alternates direction, switching back and forth in a sinusoidal pattern. In the United States, this happens 60 times per second (60 Hz), while in Europe it is 50 Hz. The voltage follows:
$$V(t) = V_0 \sin(2\pi f t)$$
where $V_0$ is the peak voltage and $f$ is the frequency. Understanding these waves is essential for electrical engineering and for designing the electronic devices you use every day.
Ferris Wheel Height
As a Ferris wheel rotates at constant speed, your height above the ground varies sinusoidally. If the wheel has radius $r$, the center is at height $h$ above the ground, and it completes one revolution in time $T$, then your height is:
$$H(t) = r\sin\left(\frac{2\pi}{T}t - \frac{\pi}{2}\right) + h$$
(The phase shift accounts for starting at the bottom of the wheel.)
This is a perfect example of circular motion producing sinusoidal behavior: as you go around the circle, your vertical position traces out a sine wave over time.
Self-Test Problems
Problem 1: What is the amplitude and period of $y = 5\cos(3x)$?
Show Answer
Amplitude: $|A| = |5| = 5$
Period: $\frac{2\pi}{|B|} = \frac{2\pi}{3}$
The wave oscillates between -5 and 5, completing one cycle every $\frac{2\pi}{3}$ units.
Problem 2: The function $y = \sin(x - \frac{\pi}{3})$ is shifted in which direction and by how much?
Show Answer
Comparing with $y = \sin(Bx - C)$ where $B = 1$ and $C = \frac{\pi}{3}$:
Phase shift $= \frac{C}{B} = \frac{\pi/3}{1} = \frac{\pi}{3}$
Since this is positive, the wave shifts $\frac{\pi}{3}$ units to the right.
Problem 3: Find the midline, maximum, and minimum values of $y = 4\sin(2x) - 3$.
Show Answer
Here $A = 4$ and $D = -3$.
Midline: $y = D = -3$
Maximum: $D + |A| = -3 + 4 = 1$
Minimum: $D - |A| = -3 - 4 = -7$
The wave oscillates between -7 and 1, centered on $y = -3$.
Problem 4: Write an equation for a cosine function with amplitude 2, period $\frac{\pi}{2}$, no phase shift, and midline $y = 5$.
Show Answer
Find A: Amplitude $= |A| = 2$, so $A = 2$ (or $-2$).
Find B: Period $= \frac{2\pi}{|B|} = \frac{\pi}{2}$
Solving: $|B| = \frac{2\pi}{\pi/2} = \frac{2\pi \cdot 2}{\pi} = 4$, so $B = 4$.
Find D: Midline is $y = 5$, so $D = 5$.
No phase shift: $C = 0$.
Equation: $y = 2\cos(4x) + 5$
Problem 5: Where are the vertical asymptotes of $y = \tan(x)$ located in the interval $[0, 2\pi]$?
Show Answer
The tangent function has asymptotes wherever $\cos x = 0$.
In $[0, 2\pi]$, $\cos x = 0$ at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$.
Asymptotes: $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$
Problem 6: A sinusoidal function has a maximum of 7 at $x = 0$ and a minimum of 1 at $x = 2\pi$. Find its equation.
Show Answer
Find amplitude and midline:
Midline $= \frac{7 + 1}{2} = 4$, so $D = 4$.
Amplitude $= 7 - 4 = 3$, so $|A| = 3$.
Find period:
The distance from max to min is half a period. Here that distance is $2\pi - 0 = 2\pi$.
So the full period is $2 \times 2\pi = 4\pi$.
$\frac{2\pi}{|B|} = 4\pi \Rightarrow |B| = \frac{1}{2}$
Find the equation:
Since the maximum is at $x = 0$, we can use $\cos$ with no phase shift:
$y = 3\cos\left(\frac{1}{2}x\right) + 4$
Verify: At $x = 0$: $y = 3\cos(0) + 4 = 3(1) + 4 = 7$ ✓
At $x = 2\pi$: $y = 3\cos(\pi) + 4 = 3(-1) + 4 = 1$ ✓
Problem 7: What is the range of $y = -3\sin(x) + 2$?
Show Answer
Here $A = -3$ and $D = 2$.
Range: $[D - |A|, D + |A|] = [2 - 3, 2 + 3] = [-1, 5]$
Note: The negative sign on $A$ reflects the wave, but does not change the range. The amplitude is still $|{-3}| = 3$.
Summary
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Sine and cosine can be defined using the unit circle: $\cos \theta$ is the x-coordinate and $\sin \theta$ is the y-coordinate of a point on the circle at angle $\theta$. This definition works for any real number, not just acute angles.
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The graph of $y = \sin x$ starts at the origin, rises to 1 at $x = \frac{\pi}{2}$, returns to 0 at $x = \pi$, drops to -1 at $x = \frac{3\pi}{2}$, and returns to 0 at $x = 2\pi$. Then it repeats.
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The graph of $y = \cos x$ starts at its maximum (1 at $x = 0$), drops to 0 at $x = \frac{\pi}{2}$, reaches its minimum (-1) at $x = \pi$, returns to 0 at $x = \frac{3\pi}{2}$, and returns to its maximum at $x = 2\pi$.
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Key features of sinusoidal functions:
- Amplitude: half the distance from max to min, equals $|A|$
- Period: the horizontal length of one complete cycle, equals $\frac{2\pi}{|B|}$
- Phase shift: the horizontal displacement, equals $\frac{C}{B}$
- Midline: the horizontal line at $y = D$, halfway between max and min
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The general sinusoidal form $y = A\sin(Bx - C) + D$ (or with cosine) allows you to describe any sinusoidal wave by adjusting four parameters.
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The tangent function $y = \tan x$ has a completely different character: it has vertical asymptotes where cosine is zero (at $x = \frac{\pi}{2} + n\pi$), period $\pi$ instead of $2\pi$, and range of all real numbers.
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Domain and range:
- Sine and cosine have domain of all real numbers and range $[-1, 1]$ (or $[D - |A|, D + |A|]$ when transformed)
- Tangent has domain excluding its asymptotes and range of all real numbers
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Sinusoidal functions model countless real-world phenomena: sound waves, ocean tides, daylight hours, electrical current, circular motion, and anything else that repeats in a smooth, periodic pattern.
Understanding these graphs gives you a visual language for describing periodic phenomena. Whether you are analyzing music, predicting tides, designing circuits, or simply appreciating the mathematical elegance of waves, the sine and cosine functions are your essential tools.