Trigonometry Graphs
When we graph the sine, cosine, and tangent functions, we uncover beautiful repeating patterns called waves. These periodic functions describe everything from sound waves to planetary motion.
This guide covers the shape and properties of each function, amplitude, period, phase shift, vertical shift, transformations, worked examples with full steps, and an interactive graph explorer.
1Introduction
While trigonometry often starts with triangles, its functions extend far beyond, describing a myriad of natural phenomena. When we graph these functions, we uncover periodic patterns — patterns that repeat indefinitely over a fixed interval.
The three primary trig graphs — sine, cosine, and tangent — each have a distinctive shape, and understanding their properties is key to working with them.
Imagine a Ferris wheel. As a specific seat goes around, its height above the ground traces out a sine or cosine wave. The highest point is the maximum, the lowest is the minimum, and the time for one full rotation is the period.
Trig graphs model sound waves, light waves, AC electricity, tides, seasons, pendulum motion, and even heartbeats. They are fundamental tools in physics, engineering, and biology.
2Key Definitions
Periodic Function
A function whose values repeat over a fixed interval: f(x + P) = f(x).
Cycle
One complete repetition of the pattern of a periodic function.
Amplitude (|A|)
Half the distance between the maximum and minimum values. Always positive: (Max − Min) / 2.
Period (P)
The horizontal length of one complete cycle. P = 2π/|B| for sin/cos, P = π/|B| for tan.
Midline (y = D)
The horizontal line halfway between max and min. Represents the vertical shift.
Phase Shift (C)
Horizontal translation. In y = A sin(B(x − C)) + D, C shifts right if positive, left if negative.
Vertical Shift (D)
Vertical translation. D > 0 shifts up, D < 0 shifts down. Sets the midline at y = D.
Asymptote
A line the graph approaches but never touches. Tangent has vertical asymptotes where cos(x) = 0.
General Form
y = A sin(B(x − C)) + D
3Sine Graph: y = sin(x)
The sine function traces the y-coordinate of a point on the unit circle as the angle increases. Its graph is a smooth, continuous wave.
Key Points (One Cycle: 0 to 2π)
| x | sin(x) | Description |
|---|---|---|
| 0 | 0 | Starts at midline, increasing |
| π/2 | 1 | Maximum |
| π | 0 | Crosses midline, decreasing |
| 3π/2 | −1 | Minimum |
| 2π | 0 | Returns to midline, one cycle complete |
Domain: All real numbers (−∞, ∞)
Range: [−1, 1]
Amplitude: 1
Period: 2π
4Cosine Graph: y = cos(x)
The cosine function traces the x-coordinate of a point on the unit circle. It has the same shape as sine but starts at its maximum value.
Key Points (One Cycle: 0 to 2π)
| x | cos(x) | Description |
|---|---|---|
| 0 | 1 | Starts at maximum |
| π/2 | 0 | Crosses midline, decreasing |
| π | −1 | Minimum |
| 3π/2 | 0 | Crosses midline, increasing |
| 2π | 1 | Returns to maximum |
Domain: All real numbers (−∞, ∞)
Range: [−1, 1]
Amplitude: 1
Period: 2π
cos(x) = sin(x + π/2). The cosine graph is just the sine graph shifted π/2 to the left. They have the same shape, amplitude, and period.
5Tangent Graph: y = tan(x)
The tangent function is defined as tan(x) = sin(x)/cos(x). Unlike sine and cosine, it has no maximum or minimum — it extends infinitely in both directions, with vertical asymptotes where cos(x) = 0.
Domain: All reals except x = π/2 + nπ
Range: All real numbers (−∞, ∞)
Amplitude: None (unbounded)
Period: π
Asymptotes occur at x = π/2 + nπ (where cos(x) = 0). The graph approaches these lines but never touches them. The x-intercepts of tangent are at x = nπ (where sin(x) = 0).
Trig Graph Explorer
InteractiveAdjust the sliders to see how amplitude, period, phase shift, and vertical shift transform the graph in real-time.
Amplitude
1
Period
6.28
Phase Shift
None
Vertical Shift
None
6Transformations
The general form y = A sin(B(x − C)) + D shows how four parameters transform the basic graph:
Amplitude (vertical stretch/compression)
|A| > 1 stretches vertically. 0 < |A| < 1 compresses. Negative A reflects across the midline.
Frequency (horizontal stretch/compression)
Period = 2π/|B| for sin/cos, π/|B| for tan. |B| > 1 compresses. 0 < |B| < 1 stretches.
Phase Shift (horizontal translation)
C > 0 shifts right. C < 0 shifts left. Must factor B out first to identify C correctly.
Vertical Shift (vertical translation)
D > 0 shifts up. D < 0 shifts down. The midline moves to y = D.
In y = sin(2x − π), the phase shift is NOT π. You must factor: y = sin(2(x − π/2)). So C = π/2, not π. Always factor B out of the parentheses first.
7Worked Examples
Problem: Find the amplitude of y = 3sin(x).
Step 1: Identify the coefficient of sin — it is 3
Step 2: Amplitude = |coefficient| = |3|
Step 3: Amplitude = 3
Answer: Amplitude = 3
Method: Amplitude formula |A|
Problem: Find the period of y = sin(2x).
Step 1: For y = sin(Bx), period = 2π/|B|
Step 2: B = 2
Step 3: Period = 2π/2 = π
Answer: Period = π
Method: Period formula 2π/|B|
Problem: Find the amplitude, period, and phase shift of y = 2cos(x − π/2).
Step 1: A = 2 → Amplitude = |2| = 2
Step 2: B = 1 (coefficient of x) → Period = 2π/1 = 2π
Step 3: (x − C) = (x − π/2) → C = π/2
Step 4: C > 0, so graph shifts π/2 to the right
Answer: Amplitude = 2, Period = 2π, Phase Shift = π/2 right
Method: Transformation parameters A, B, C
Problem: A sine wave has amplitude 1 and period 4π. Write its equation.
Step 1: Amplitude = 1, so A = 1
Step 2: Period = 4π = 2π/|B|
Step 3: |B| = 2π/(4π) = 1/2
Step 4: B = 1/2
Step 5: Equation: y = sin(x/2)
Answer: y = sin(x/2)
Method: Deriving equation from amplitude and period
Problem: Describe the graph of y = tan(x/2).
Step 1: Tangent has no amplitude (unbounded)
Step 2: B = 1/2 → Period = π/|B| = π/(1/2) = 2π
Step 3: Asymptotes where Bx = π/2 + nπ
Step 4: x/2 = π/2 + nπ → x = π + 2nπ
Step 5: Asymptotes at x = …, −π, π, 3π, …
Step 6: Passes through the origin (0, 0)
Answer: Period = 2π, asymptotes at x = π + 2nπ
Method: Tangent period and asymptote formulas
Problem: Identify all parameters of y = −3sin(2x − π) + 1.
Step 1: Factor B out: y = −3sin(2(x − π/2)) + 1
Step 2: A = −3 → Amplitude = |−3| = 3, reflected
Step 3: B = 2 → Period = 2π/2 = π
Step 4: C = π/2 → Phase shift = π/2 right
Step 5: D = 1 → Vertical shift = 1 up, midline y = 1
Answer: Amp = 3 (reflected), Period = π, Shift = π/2 right, Midline y = 1
Method: Factor B first, then identify A, B, C, D
8Memory Aids
sin(0) = 0 (starts at midline). cos(0) = 1 (starts at maximum). This is the key difference between the two graphs.
Sine and cosine complete a full cycle in 2π/|B|. Tangent completes its cycle in half that: π/|B|.
For phase shift: (x − C) shifts right. (x + C) shifts left. It’s counter-intuitive — minus goes right!
Before identifying the phase shift, always rewrite as sin(B(x − C)). Don’t read C directly from sin(Bx + something).
Amplitude = |A|, always a positive distance. A negative A doesn’t make the amplitude negative — it reflects the graph.
9Common Mistakes to Avoid
Saying Amplitude is Negative
Wrong: “The amplitude of y = −2sin(x) is −2.”
Right: Amplitude = |−2| = 2. The negative sign causes a reflection, not a negative amplitude. Amplitude is always positive.
Using Wrong Period Formula
Wrong: Period of sin(2x) = 2π × 2 = 4π
Right: Period = 2π/|B| = 2π/2 = π. Divide by B, don’t multiply. For tangent, use π/|B|.
Confusing Phase Shift Direction
Wrong: sin(x + π) shifts right by π.
Right: sin(x + π) = sin(x − (−π)) shifts left by π. Remember: minus moves right, plus moves left.
Mixing Up Sine and Cosine Starting Points
Wrong: Drawing cosine starting at (0, 0) like sine.
Right: Sine starts at the midline (0, 0). Cosine starts at its maximum (0, 1). They have the same shape but different starting points.
Wrong Tangent Asymptote Locations
Wrong: Asymptotes of tan(x) at x = nπ.
Right: Asymptotes at x = π/2 + nπ (where cos = 0). x = nπ are the x-intercepts (where sin = 0).
Not Factoring B Before Reading Phase Shift
Wrong: Phase shift of sin(2x − π) is π.
Right: Factor first: sin(2(x − π/2)). The phase shift is π/2, not π. Always factor B out of the argument.
Assuming Tangent Has an Amplitude
Wrong: “The amplitude of y = 2tan(x) is 2.”
Right: Tangent has no amplitude because its range is all real numbers (−∞ to ∞). The coefficient 2 is a vertical stretch, not an amplitude.
Confusing Period of Sine/Cosine vs Tangent
Wrong: Using 2π/|B| for the tangent period.
Right: Sine/cosine: period = 2π/|B|. Tangent: period = π/|B|. Tangent’s period is half that of sine and cosine.
10Quick Revision Summary
- Sine starts at the midline (0, 0). Cosine starts at its maximum (0, 1).
- Tangent has no amplitude and has vertical asymptotes at π/2 + nπ.
- Amplitude = |A| (always positive). Negative A causes reflection.
- Period = 2π/|B| for sine/cosine, π/|B| for tangent.
- Phase shift: (x − C) shifts right, (x + C) shifts left.
- Vertical shift D moves the midline to y = D.
- Always factor B out before identifying phase shift C.
- cos(x) = sin(x + π/2) — cosine is sine shifted left by π/2.
- Sine/cosine range: [−|A| + D, |A| + D]. Tangent range: all reals.
- Use the interactive explorer above to build intuition for each transformation.
Frequently Asked Questions
What is the difference between the sine and cosine graph?
The sine graph starts at the origin (0, 0) and goes up first, while the cosine graph starts at its maximum (0, 1) and goes down first. Cosine is essentially a sine graph shifted π/2 to the left: cos(x) = sin(x + π/2).
Why does the tangent graph have vertical asymptotes?
Because tan(x) = sin(x)/cos(x), and division by zero is undefined. Wherever cos(x) = 0 (at x = π/2 + nπ), the tangent function is undefined, creating vertical asymptotes.
How do I find the period of a trig function?
For y = A sin(Bx) or y = A cos(Bx), the period is 2π/|B|. For y = A tan(Bx), the period is π/|B|. The value B is the coefficient of x inside the function.
Can amplitude be negative?
No. Amplitude is always positive because it represents a distance (half the height of the wave). If the coefficient A is negative, the amplitude is |A|, and the negative sign causes a reflection over the midline.
What real-world phenomena use trig graphs?
Sound waves, light waves, AC electricity, tides, seasons, pendulum motion, Ferris wheels, radio signals, and heartbeats all follow sinusoidal patterns that can be modeled with trig graphs.
Practice Quiz
Test your understanding — select the correct answer for each question.
1.What is the amplitude of y = 4sin(x)?
2.What is the period of y = sin(3x)?
3.Where does y = cos(x) start (at x = 0)?
4.What happens to the sine graph with sin(x − π)?
5.What is the period of y = tan(x)?
6.The midline of a sinusoidal function is:
7.y = sin(x + 2) shifts:
8.What is the domain of y = tan(x)?
9.For y = −cos(x), the negative indicates:
10.What is the range of y = 3sin(x)?
Study Tips
- Sketch the basic shapes first — know what sin, cos, and tan look like before applying transformations.
- Use the interactive explorer above — adjust amplitude, frequency, phase shift, and vertical shift to see each effect individually.
- Factor B out before identifying phase shift — this is the most common source of errors on tests.
- Remember the starting points — sin starts at 0, cos starts at max. This helps you sketch transformed graphs quickly.
- Practice writing equations from graphs — read off amplitude, period, shifts, then assemble A, B, C, D.
Related Topics
The Unit Circle
Special angles, coordinates, reference angles, and the foundation of trig functions
Right Triangle Trigonometry
SOH-CAH-TOA, finding sides and angles, inverse trig functions
Functions and Graphs
Function notation, types, transformations, and graphing
Quadratic Equations
Solving, graphing, and applying quadratic equations