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
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
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.
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.
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:
• Unit Circle — the geometric source of every trig function value. • Angle Explorer — visualize angles, quadrants, reference angles, and related-angle relationships. • Trigonometric Identities — Pythagorean, reciprocal, quotient, and angle-sum formulas. • Inverse Trigonometric Functions — arcsin, arccos, arctan and their restricted domains. • Trig Equations Solver — practice solving equations involving sine, cosine, and tangent. • Special Angles Table — exact values at 0°, 30°, 45°, 60°, 90°, and beyond.
The Graph of Sine
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.