Trig Graphs Calculator
Interactively graph sine, cosine, tangent, cotangent, secant, and cosecant trig graphs with real-time control of amplitude, period, phase shift, and vertical shift. Graphing trigonometric functions has never been more visual — every key feature calculated instantly as you drag.
Quick Examples:
All six trig graphs displayed simultaneously — click any card to load it into the Graph Builder for interactive exploration.
Overlay up to 4 trig graphs simultaneously. Expand each function's controls to adjust A, B, C, D individually.
Type any trig equation in the form A·func(Bx+C)+D and instantly extract all key features including amplitude, period, and phase shift.
Try these examples:
The Six Trig Function Graphs — Key Properties
Understanding trig graphs begins with the six fundamental trigonometric graphs: sine, cosine, tangent, cotangent, secant, and cosecant. Each has a distinctive shape, period, and range that defines its behavior. Graphing trigonometric functions systematically requires knowing these properties before applying any transformations.
Sine and cosine are the foundational trig graphs — both produce smooth, continuous S-shaped waves with period 2π and amplitude 1, bounded between −1 and 1. The key relationship: sin(x + π/2) = cos(x) — the cosine graph is simply the sine graph shifted π/2 units to the left. This co-function identity means you only need to master one wave shape for both.
Tangent and cotangent have period π (not 2π — one of the most common mistakes in graphing trig functions) and are unbounded, meaning their range is all real numbers. Vertical asymptotes divide each into separate branches. Secant and cosecant are reciprocals of cosine and sine respectively, producing U-shaped branches with range (−∞,−1] ∪ [1,+∞).
Graphing y = A·sin(Bx + C) + D — The Four Transformations
Every trig graph transformation of the form y = A·sin(Bx + C) + D (or y = A·cos(Bx + C) + D, etc.) involves exactly four parameters. Mastering these four transformations is the complete skill of graphing trigonometric functions.
A — Amplitude (Vertical Stretch)
Amplitude = |A|. This is the distance from the midline to the maximum or minimum. When A = 2, the graph reaches y = 2 and y = −2 instead of y = 1 and y = −1. When A is negative, the graph reflects over the x-axis (flips upside down) — but amplitude is still |A|, always positive. For y = A·sin(Bx+C)+D the amplitude transformation applies only to sin, cos (and by extension sec, csc). Tan and cot have no defined amplitude.
B — Frequency / Period Multiplier
New period = 2π / |B| for sin, cos, sec, csc. New period = π / |B| for tan, cot. When B = 2, the period of sin halves to π — the graph completes two full cycles in the same space. When B = 1/2, the period doubles to 4π. When B is negative, the graph reflects horizontally. Never confuse period with frequency: period = 2π/|B|, not 2π × |B|.
C — Phase Shift (Horizontal Shift)
Phase shift = −C/B. When phase shift is positive, the graph moves RIGHT; negative means LEFT. The most common confusion: in y = sin(Bx + C), adding C (positive C) shifts the graph LEFT because the function reaches its "zero" when Bx + C = 0, i.e., when x = −C/B. This is why sin(x + π/4) shifts LEFT by π/4 and sin(x − π/4) shifts RIGHT by π/4.
Critical: sin(x − π/2) = cos(x). A phase shift of π/2 to the RIGHT converts sine into cosine. This is one of the most important co-function relationships in trigonometry and demonstrates that sin and cos are the same trig graph with a different starting point.
D — Vertical Shift (Midline)
Midline: y = D. The entire graph moves up by D (if D > 0) or down (if D < 0). The maximum becomes |A| + D and the minimum becomes −|A| + D. The midline y = D replaces the x-axis as the center of oscillation. For graphing trigonometric functions, always draw the midline as a dashed reference line before plotting the wave.
How to Graph Trig Functions — Step-by-Step Method
Use this five-step method for graphing trig functions by hand. It works for any trig graph of the form y = A·sin(Bx+C)+D.
The 5-Step Method for Graphing Trigonometric Functions
- Find Amplitude: |A|. Note if A < 0 (reflection over x-axis).
- Find Period: 2π/|B| for sin/cos/sec/csc, or π/|B| for tan/cot.
- Find Phase Shift: −C/B. Positive = right, negative = left.
- Find Midline: y = D. Draw as a dashed horizontal reference line.
- Plot 5 key points per period starting from the phase shift: at x₀, x₀ + T/4, x₀ + T/2, x₀ + 3T/4, x₀ + T (where T = period). Connect smoothly.
Worked Example: Graph y = 2sin(3x − π/2) + 1
- A = 2, so amplitude = 2 (no reflection since A > 0)
- B = 3, so period = 2π/3 ≈ 2.094
- C = −π/2, so phase shift = −(−π/2)/3 = π/6 to the RIGHT
- D = 1, so midline is y = 1; maximum = 2+1 = 3; minimum = −2+1 = −1
- Starting x = π/6. Key points: (π/6, 1), (π/6+π/6, 3), (π/6+π/3, 1), (π/6+π/2, −1), (π/6+2π/3, 1)
Graphing Sine and Cosine — The Basic Waves
The sin graph and cos graph are the two most fundamental trig graphs. They have identical shapes — both are smooth sinusoidal waves with period 2π, amplitude 1, and range [−1, 1]. The only difference is their starting position: cos starts at its maximum (0, 1) while sin starts at its zero (0, 0).
| Property | sin(x) | cos(x) |
|---|---|---|
| Period | 2π | 2π |
| Amplitude | 1 | 1 |
| Range | [−1, 1] | [−1, 1] |
| Zeros | 0, π, 2π, … | π/2, 3π/2, … |
| Maximum (y=1) | x = π/2 + 2nπ | x = 2nπ |
| Minimum (y=−1) | x = 3π/2 + 2nπ | x = π + 2nπ |
| Symmetry | Odd: sin(−x) = −sin(x) | Even: cos(−x) = cos(x) |
The 5 key points of sin(x): (0,0) → (π/2, 1) → (π, 0) → (3π/2, −1) → (2π, 0)
The 5 key points of cos(x): (0,1) → (π/2, 0) → (π, −1) → (3π/2, 0) → (2π, 1)
Graphing Tangent and Cotangent — Period π and Asymptotes
The tan graph is fundamentally different from sin and cos. Its period is π — half that of sine and cosine. It is unbounded (range = all real numbers) with vertical asymptotes at x = π/2 + nπ where cos(x) = 0. Between consecutive asymptotes, tan increases from −∞ to +∞, passing through zero.
For graphing trig functions like tan and cot, always draw the asymptotes first as vertical dashed red lines, then sketch the curve between them. For tan(x), the curve passes through (0,0), rises steeply toward π/2, and falls from −∞ just after −π/2. Cotangent is the "mirror image" — it decreases on each branch, with asymptotes at x = nπ and zeros at π/2 + nπ.
Key differences between tan and cot graphs:
- tan(x): asymptotes at x = π/2 + nπ, zeros at nπ, INCREASING on each branch
- cot(x): asymptotes at x = nπ, zeros at π/2 + nπ, DECREASING on each branch
- Period of both: π (not 2π — the most common mistake in graphing trig functions)
- cot(x) = cos(x)/sin(x): related to tan by cot(x) = 1/tan(x)
Graphing Secant and Cosecant — Reciprocal Functions
Secant and cosecant are the most challenging trig graphs to sketch. The standard textbook technique for graphing circular functions like sec and csc is to first draw the parent function (cos or sin) as a dashed guide, then draw the reciprocal branches.
Key rule: Where the parent function touches y = 1, the reciprocal function also equals 1. Where the parent touches y = −1, the reciprocal also equals −1. Where the parent crosses zero, the reciprocal has a vertical asymptote. The U-shaped branches of sec(x) open upward where cos(x) > 0 and downward where cos(x) < 0.
How to Graph sec(x) from cos(x):
- Draw y = cos(x) as a dashed guideline
- Mark the zeros of cos(x) at x = π/2 + nπ — these become vertical asymptotes for sec(x)
- Mark the maxima of cos(x) at x = 2nπ — sec(x) also touches these points (y=1)
- Mark the minima of cos(x) at x = π+2nπ — sec(x) also touches (y=−1)
- Draw U-shaped branches between consecutive asymptotes, touching the cos curve at its peaks/troughs
Phase Shift — The Most Confusing Transformation
The phase shift direction is the single most common source of errors when graphing trigonometric functions. Students consistently shift the graph in the wrong direction. Here is the definitive explanation.
Phase Shift Direction Rules
shifts LEFT by C/B
Adding to x shifts LEFT
shifts RIGHT by C/B
Subtracting from x shifts RIGHT
Phase shift = −C/B | Positive result = shift RIGHT | Negative result = shift LEFT
Three Phase Shift Worked Examples
- sin(x + π/4): C=π/4, B=1 → phase shift = −π/4 → shifts LEFT π/4 ✓
- sin(x − π/4): C=−π/4, B=1 → phase shift = +π/4 → shifts RIGHT π/4 ✓
- sin(2x − π): C=−π, B=2 → phase shift = −(−π)/2 = π/2 → shifts RIGHT π/2 (NOT π — divide by B!) ✓
Common mistake: For sin(2x+π), students say phase shift = π left. Wrong — it's π/B = π/2 left. Always divide C by B.
Common Mistakes When Graphing Trig Functions
Mistake 1 — Period Confusion: confusing period with B
- ❌ Wrong: period of sin(3x) = 3 × 2π = 6π
- ✅ Correct: period = 2π/|B| = 2π/3 ≈ 2.09
Mistake 2 — Phase Shift Direction Error
- ❌ Wrong: sin(x + π/4) shifts RIGHT π/4
- ✅ Correct: sin(x + π/4) shifts LEFT π/4 (adding to x shifts left)
Mistake 3 — Not Factoring B Before Finding Phase Shift
- ❌ Wrong: sin(2x + π) has phase shift π to the left
- ✅ Correct: sin(2x+π) = sin(2(x+π/2)) → phase shift = π/2 to the LEFT
Mistake 4 — Drawing sec/csc Without Parent Guide First
- ❌ Wrong: attempting sec(x) without first drawing cos(x) as reference
- ✅ Correct: always draw parent sin/cos first, then draw reciprocal branches at peaks/troughs
Mistake 5 — Drawing tan(x) with Period 2π
- ❌ Wrong: period of tan(x) = 2π (same as sin/cos)
- ✅ Correct: period of tan(x) = π — tan repeats every π units, not every 2π
Trig Graph Reference — All Six Functions Side by Side
Complete reference table for all six trigonometric graphs. This is the essential guide for graphs of trig functions — all properties for all six functions in one place for quick reference when graphing trig functions.
| Function | Equation | Period | Range | Zeros | Asymptotes | Symmetry |
|---|---|---|---|---|---|---|
| sin | y=sin(x) | 2π | [−1, 1] | nπ | None | Odd |
| cos | y=cos(x) | 2π | [−1, 1] | π/2+nπ | None | Even |
| tan | y=tan(x) | π | (−∞,+∞) | nπ | π/2+nπ | Odd |
| cot | y=cot(x) | π | (−∞,+∞) | π/2+nπ | nπ | Odd |
| sec | y=sec(x) | 2π | (−∞,−1]∪[1,+∞) | None | π/2+nπ | Even |
| csc | y=csc(x) | 2π | (−∞,−1]∪[1,+∞) | None | nπ | Odd |
Worked Examples — Full Graphing Problems
1. Graph y = 3cos(2x) — Find all key features
- A=3: amplitude=3, no reflection
- B=2: period=2π/2=π
- C=0: phase shift=0 (no shift)
- D=0: midline y=0, max=3, min=−3
- Key points per period: (0,3),(π/4,0),(π/2,−3),(3π/4,0),(π,3)
2. Graph y = −2sin(x + π/3) — note the negative amplitude and phase shift
- A=−2: amplitude=2, REFLECTED over x-axis (flipped)
- B=1: period=2π
- C=π/3: phase shift=−π/3 → shifted LEFT by π/3
- D=0: midline y=0, max=2, min=−2
- Note: because of reflection, starts going DOWN from zero instead of up
3. Graph y = tan(x/2) — period doubles
- B=1/2: period=π/(1/2)=2π — period of this tan graph is 2π not π
- Asymptotes at x = π + 2nπ (solving (1/2)x = π/2 + nπ → x = π + 2nπ)
- Zeros at x = 2nπ
- Sketch: vertical asymptotes at x=…−π, π, 3π…, curve passes through (0,0)
4. Graph y = 2sin(3x − π/2) + 1 — full transformation
- A=2: amplitude=2
- B=3: period=2π/3
- C=−π/2: phase shift = −(−π/2)/3 = π/6 RIGHT
- D=1: midline y=1, max=3, min=−1
- Note: sin(x − π/2) = −cos(x), so y=2sin(3x−π/2)+1 is related to −cos
5. Prove sin(x + π/2) = cos(x) using the phase shift
- C=π/2, B=1: phase shift = −π/2 → graph shifts LEFT π/2
- When sin graph moves LEFT by π/2, it starts at its maximum → becomes cos(x)
- Algebraic proof: sin(x+π/2) = sin(x)cos(π/2)+cos(x)sin(π/2) = 0+cos(x) = cos(x) ✓
6. Graph y = sec(x) from y = cos(x)
- Draw cos(x) dashed: period=2π, maxima at 2nπ, minima at π+2nπ, zeros at π/2+nπ
- Draw vertical asymptotes at each zero of cos(x): x = π/2 + nπ
- Draw U-branches opening UP (touching cos peaks at y=1) and DOWN (touching troughs at y=−1)
- sec graph never enters the band −1 < y < 1
7. Find equation from graph: period=4, amplitude=3, shifts up 2, starts at zero going down
- Amplitude A=−3 (negative because going DOWN from zero → reflected sin)
- Period=4 → 2π/B=4 → B=π/2
- D=2 (vertical shift up 2)
- Equation: y = −3sin((π/2)x) + 2
8. Graph y = 2csc(x) + 1 from y = 2sin(x) + 1
- Draw parent: y=2sin(x)+1, period=2π, max=3, min=−1
- Asymptotes for csc at zeros of sin(x): x=nπ
- Branches touch parent at maxima (y=3) and minima (y=−1)
- csc graph stays above y=1+2=3 or below y=1−2=−1 — never between