The Function row switches the curve between sin, cos, tan, csc, sec and cot. The angle stays where it is, so the new curve appears with the marker at the same θ. Learn more about selecting a function
The Unit buttons switch every angle on the page between degrees and radians: the axis labels, the angle box, the preset buttons and the result. Learn more about degrees and radians
Drag the Angle slider, or type an exact value in the box beside it, to move the marker along the curve from −360° to 360°. Learn more about setting the angle
The Quick buttons jump straight to the special angles 0, 6π, 4π, 3π, 2π, π, 23π and 2π. Learn more about the presets
The Result field reads the current value, such as sin(2π)=1.0000, and says undefined where the function has no value. Learn more about the result display
On the graph, the red marker sits at the current angle and dashed lines mark asymptotes. Drag the plot to pan, scroll or use the x± and y± buttons to zoom, and Reset to restore the view. Learn more about reading the graph
Hold the angle fixed and click through the functions to compare their values at one point — the fastest way to see reciprocal and quotient relationships. Learn more about comparing functions
• Period — the smallest positive value T such that f(θ+T)=f(θ) for all θ. sin, cos, sec, csc have period2π; tan and cot have period π. • Amplitude — half the distance between the maximum and minimum values. Defined for bounded functions only (sin and cos have amplitude1). • Range — the set of output values. sin and cos: [−1,1]. sec and csc: (−∞,−1]∪[1,∞). tan and cot: all real numbers. • Asymptote — a vertical line the graph approaches but never crosses. Occurs where the function is undefined (denominator equals zero). • Zero — input value where the function equals zero. The x-axis intercepts of the curve.
What changes when you switch: • The plotted curve redraws with the function's characteristic shape. • The result expression updates to use the new function name. • The explanation panel on the right replaces its text with definitions and key features for the chosen function.
The selected function stays highlighted in dark blue, making it easy to track which curve you are viewing during a comparison session.
Switching Between Degrees and Radians
The Unit toggle switches the x-axis labeling and the angle input between deg and rad.
How the conversion works: • The underlying angle is stored internally and does not change when you switch units. • In deg mode, presets show 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°. • In rad mode, presets show 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π.
Use radians when working with calculus, periodicity proofs, or the unit circle, and degrees when the problem comes from geometry or applied measurement.
Setting the Angle
DemoAngle, units and presets
Step 0 of 5
Three input methods control the current angle, all linked together:
• Slider — drag horizontally for a continuous sweep from −360° to +360° in steps of 1°. • Number input — type an exact value. The unit symbol next to the box reflects the active unit. • Quick presets — click any of the eight buttons to jump to a special angle.
Watching the result value update as you drag the slider builds strong intuition for how each function changes with θ. Try dragging through a full period for sin to see the wave shape emerge.
Reading the Result Display
DemoResult and comparison
Step 0 of 5
The Result field in the upper-right of the controls panel shows the current evaluation in the form:
f(θ)=value
For example, with sin selected and the angle at π/2, the display reads sin(π/2)=1.0000.
Key behaviors: • Values are rounded to four decimal places. • Radian angles are formatted as fractions of π when possible (e.g., π/4, 3π/2). • When the function is mathematically undefined at the chosen angle, the display reads "undefined" instead of a number.
This makes the explorer useful as a quick lookup tool while still showing the underlying graph context.
Reading the Graph
The graph plots the selected function across a range of angle values, with a vertical marker showing your current θ.
What to look for: • Wave shape — sin and cos trace smooth bounded oscillations between −1 and 1. • Asymptotes — tan, cot, sec, csc shoot to ±∞ at the angles where their denominators vanish. • Periodicity — the same pattern repeats. Compare θ and θ+2π to confirm. • Zeros — points where the curve crosses the x-axis.
Sliding the angle input animates the marker along the curve, anchoring numerical output to its visual position. Drag the plot sideways to pan the visible range, and use the mouse wheel over it — or the x+, x−, y+ and y− buttons above it — to zoom in or out; Reset returns to the default window.
Comparing Functions Side by Side
Although only one function is shown at a time, you can build a mental overlay quickly:
• Pick a fixed angle, then click each function button in sequence and note the result values. • Use π/4 (45°) to see that sin and cos both equal 2/2, while tan equals 1. • At π/2 (90°), sin peaks at 1, cos hits zero, and tan becomes undefined. • Comparing sin with csc (or cos with sec) at the same angle highlights the reciprocal relationship: their product equals 1 wherever both are defined.
This pattern of stepping through functions at fixed angles is one of the fastest ways to internalize trig identities.
The Explanations Panel
The Explanations card on the right of the tool always describes the function that is currently selected. Switching the Function row rewrites it: sine and cosine are described as coordinates on the unit circle, tangent and cotangent as ratios with their asymptotes, and the three reciprocals by the function they invert.
Each explanation states the facts a reader needs to read the graph: the range, the period, where the zeros are, where the maxima and minima sit, and where the function is undefined. Those are exactly the features to look for on the curve — the card and the plot are meant to be read together.
The card does not change with the angle. It describes the function as a whole; the Result field and the marker on the curve describe the current value. Keep both in view: pick a function, read what the card says about it, then drag the angle and watch the curve do what the card promised.
Each explanation ends with a link to the full treatment of that function further down this page, where the graph is frozen at 60° and every feature is annotated.
What Are Trigonometric Functions?
A trigonometric function assigns a numeric value to every angle. The three primary functions come from the unit circle: for a point at angle θ on the circle, cosθ is its x-coordinate, sinθ is its y-coordinate, and tanθ=sinθ/cosθ.
The three reciprocal functions follow directly: • cscθ=1/sinθ • secθ=1/cosθ • cotθ=1/tanθ
For full theory and definitions, see the trigonometric functions theory page.
Period, Amplitude, and Range
Three properties summarize the global behavior of each function:
• Period — how often the pattern repeats. sin, cos, sec, csc repeat every 2π; tan and cot repeat every π. • Amplitude — half the peak-to-trough distance, defined only for bounded functions. sin and cos have amplitude 1. • Range — the set of possible output values. sin and cos stay in [−1,1]; sec and csc live outside (−1,1); tan and cot cover all real numbers.
For full coverage with proofs and transformations, see the period and amplitude page.
Asymptotes and Undefined Points
Four of the six functions have vertical asymptotes — vertical lines the graph approaches but never crosses, marking inputs where the function is undefined:
• tanθ and secθ are undefined where cosθ=0, i.e., at θ=π/2+πk. • cotθ and cscθ are undefined where sinθ=0, i.e., at θ=πk.
The explorer reports "undefined" at these inputs and the curve appears to break in the graph. For a deeper look at why these gaps appear, see the trig identities page.
Related Concepts and Tools
Continue exploring with these connected resources:
Sine traces the classic bounded wave: the y-coordinate of the unit-circle point, plotted against the angle. Its graph oscillates smoothly between −1 and 1 with no breaks and no asymptotes.
y = sin θ, frozen at θ = 60°
The bounded wave between −1 and 1; the marker rides at 60°, where sin θ = √3/2 ≈ 0.866.
Key features, all visible in the frozen graph: period 2π (360°), amplitude 1, zeros at every integer multiple of π, maxima at π/2+2πk and minima at −π/2+2πk. The marker sits at θ=60°, where sinθ=3/2≈0.866.
Sine is an odd function — the graph has point symmetry through the origin — and it is the cosine curve shifted right by π/2. Its reciprocal is cosecant, whose asymptotes stand exactly at sine's zeros.
The Graph of Cosine
Cosine is the x-coordinate of the unit-circle point: the same bounded wave as sine, but starting from its maximum of 1 at θ=0.
y = cos θ, frozen at θ = 60°
The same wave started from its maximum at θ = 0; at the marker, cos θ = 1/2 exactly.
Key features: period 2π, amplitude 1, zeros at π/2+πk, maxima at 2πk, minima at π+2πk. At the frozen θ=60° the marker reads cosθ=1/2.
Cosine is an even function — mirror-symmetric about the y-axis — and equals sine shifted left by π/2. Its reciprocal is secant, which diverges precisely at cosine's zeros.
The Graph of Tangent
Tangent is the ratio sinθ/cosθ — the slope of the unit-circle ray — and its graph is a family of increasing branches separated by vertical asymptotes.
y = tan θ, frozen at θ = 60°
Climbing branches fenced by asymptotes π apart; at the marker, tan θ = √3 ≈ 1.732.
Key features: period π — half that of sine and cosine — with asymptotes at π/2+πk (where cosθ=0), zeros at πk, and a range covering all real numbers. At the frozen θ=60°, tanθ=3≈1.732.
Each branch climbs from −∞ to +∞ across its interval. The reciprocal is cotangent, whose branches fall instead of climb and whose asymptotes sit at tangent's zeros.
The Graph of Cosecant
Cosecant is 1/sinθ: a train of U-shaped branches that open away from the x-axis, riding on the peaks and valleys of the sine wave.
y = csc θ, frozen at θ = 60°
U-branches opening away from the axis, asymptotes standing at sine’s zeros; the marker reads csc θ ≈ 1.155.
Key features: period 2π, vertical asymptotes at πk (where sinθ=0), and a range of (−∞,−1]∪[1,∞) — the curve never enters the open band between −1 and 1. At the frozen θ=60°, cscθ=2/3≈1.155.
Each branch touches ±1 exactly where sine peaks, since a maximum of sine is a minimum of its reciprocal.
The Graph of Secant
Secant is 1/cosθ: the same U-branch architecture as cosecant, positioned over the cosine wave instead.
y = sec θ, frozen at θ = 60°
The same U-branch architecture positioned over the cosine wave; at the marker, sec θ = 2 exactly.
Key features: period 2π, vertical asymptotes at π/2+πk (where cosθ=0), range (−∞,−1]∪[1,∞). At the frozen θ=60°, secθ=2 exactly — the reciprocal of cos60°=1/2.
Its branches touch ±1 at cosine's extremes, and its asymptotes coincide with tangent's, since both divide by cosθ.
The Graph of Cotangent
Cotangent is cosθ/sinθ — tangent inverted — and its graph is a family of strictly decreasing branches.
y = cot θ, frozen at θ = 60°
Strictly falling branches interlocking with tangent’s; the marker reads cot θ = 1/√3 ≈ 0.577.
Key features: period π, vertical asymptotes at πk (where sinθ=0), zeros at π/2+πk, range all real numbers. At the frozen θ=60°, cotθ=1/3≈0.577.
Cotangent's asymptotes stand at tangent's zeros and vice versa — the two graphs interlock — and it shares its asymptote positions with cosecant, its partner in dividing by sinθ.