Trigonometric Function Graphs
Visualize sin, cos, tan, cot, sec, and csc with adjustable amplitude, period, and phase shift.
Interactive Trig Graph Plotter
Trigonometric Graph Properties
Trigonometric graph properties describe the visual and mathematical characteristics of each function’s curve. Understanding period, amplitude, and asymptotes allows you to quickly sketch or interpret any trig graph.
| Function | Period | Amplitude | Range | Asymptotes |
|---|---|---|---|---|
| sin(x) | 2π (360°) | 1 | [−1, 1] | None |
| cos(x) | 2π (360°) | 1 | [−1, 1] | None |
| tan(x) | π (180°) | Unlimited | (−∞, ∞) | x = π/2 + nπ |
| cot(x) | π (180°) | Unlimited | (−∞, ∞) | x = nπ |
| sec(x) | 2π (360°) | Unlimited | (−∞,−1]∪[1,∞) | x = π/2 + nπ |
| csc(x) | 2π (360°) | Unlimited | (−∞,−1]∪[1,∞) | x = nπ |
Understanding Trigonometric Graphs
Understanding trigonometric graphs starts with recognizing the key features every periodic function shares: midline, amplitude, period, and phase shift. Each of these features tells you something specific about how the curve behaves and how it has been transformed from the basic form.
Midline
The midline is the horizontal line that the function oscillates around. For standard sin and cos functions, the midline is y = 0. A vertical shift moves the midline up or down.
Amplitude
Amplitude measures the maximum distance the graph reaches above or below its midline. For A·sin(x), amplitude = |A|. It always has a positive value.
Period
The period is the horizontal length of one complete cycle. For sin(Bx) and cos(Bx), the period equals 2π divided by |B|. For tan(Bx) and cot(Bx), period equals π divided by |B|.
Phase Shift
Phase shift is horizontal displacement of the graph. For sin(x − C), the shift is C units to the right. A negative C shifts the graph to the left.
The Sine Curve
The sine curve is the most fundamental trigonometric graph. Starting at the origin (0, 0), it rises to a maximum of 1 at π/2 radians, returns through zero at π, drops to a minimum of −1 at 3π/2, and completes its cycle at 2π. This smooth, continuous wave shape appears throughout nature in sound waves, light waves, and alternating current electricity.
The Cosine Curve
The cosine curve has the same shape as sine but is shifted π/2 radians to the left, meaning it starts at its maximum value of 1 when x = 0. Cosine and sine are related by the identity cos(x) = sin(x + π/2), making them the same wave viewed from a different starting point.
Tangent and Its Asymptotes
The tangent graph looks quite different from sine and cosine. It has a period of π (180°) instead of 2π, and it has no bounded amplitude. The curve rises from negative infinity, passes through zero, and climbs to positive infinity before jumping discontinuously to the next cycle. The vertical lines where it is undefined, at x = π/2, 3π/2, and so on, are called vertical asymptotes.
Secant and Cosecant Curves
Secant and cosecant graphs are shaped like repeating U-curves that open alternately upward and downward. Because they are the reciprocals of cosine and sine respectively, they take the value +1 or −1 wherever their base function reaches its maximum or minimum, and they shoot off to infinity wherever their base function crosses zero.
Frequently Asked Questions
The sine graph is a smooth, continuous wave that starts at 0, rises to 1 at 90°, returns to 0 at 180°, falls to −1 at 270°, and returns to 0 at 360°. This pattern repeats endlessly in both directions along the x-axis.
The period of the cosine function is 2π radians, which equals 360 degrees. The cosine graph repeats every 360 degrees. If a coefficient B is placed inside the function as cos(Bx), the period becomes 2π divided by the absolute value of B.
Amplitude is the maximum height a trigonometric graph reaches above or below its midline. For A·sin(x), the amplitude equals |A|. The standard sine and cosine functions have an amplitude of 1, ranging from −1 to 1.
Tangent has vertical asymptotes because tan(x) = sin(x)/cos(x), and division by zero is undefined. At every angle where cosine equals zero (90°, 270°, and so on), the tangent function becomes undefined and the graph has a vertical asymptote.
To read a trigonometric graph, identify the midline (the center line around which the wave oscillates), the amplitude (the distance from midline to peak or trough), the period (the horizontal length of one complete cycle), and any phase shift (leftward or rightward displacement from the standard form).
The secant graph (sec) is the reciprocal of cosine, so it has vertical asymptotes where cosine equals zero, at 90° and 270°. The cosecant graph (csc) is the reciprocal of sine, so it has vertical asymptotes where sine equals zero, at 0° and 180°. Both graphs form U-shaped curves alternating above and below zero.