Plot sine, cosine, tangent and their reciprocals with real-time controls for amplitude, period, phase shift, and vertical shift.
Visualize sine, cosine, tangent and more with live sliders. See how amplitude, period, phase shift, and vertical shift transform your function in real time.
A (Amplitude) stretches the graph vertically. |A| is the maximum distance from the midline.
B (Period Factor) compresses horizontally. Period = 2π/|B| for sin and cos.
C (Phase Shift) moves the graph left or right by C/B units.
D (Vertical Shift) moves the entire graph up or down.
Trigonometric functions are periodic functions that form the backbone of countless applications in science, engineering, and mathematics. The six standard trig functions -- sine, cosine, tangent, cosecant, secant, and cotangent -- each produce a distinct graph shape. By understanding how to transform these graphs using amplitude, period, phase shift, and vertical shift, you can model almost any periodic phenomenon.
The general form of a transformed trig function is written as y = A · f(Bx - C) + D. Each parameter controls a specific transformation. Mastering these four parameters gives you complete control over the shape and position of any trigonometric graph.
The amplitude |A| determines the vertical stretch or compression of the function. For sine and cosine, it represents the maximum distance from the midline to the peak (or trough). A basic sine wave oscillates between -1 and 1; setting A = 3 makes it oscillate between -3 and 3. If A is negative, the graph is reflected vertically (flipped upside down). Note that tangent, cosecant, secant, and cotangent do not have a bounded amplitude since they extend to infinity.
The B value controls horizontal compression or stretching. Larger values of B compress the graph horizontally, producing more cycles in the same interval. For example, B = 2 doubles the frequency, halving the period from 2π to π. This parameter directly relates to the physical concept of frequency in sound waves and oscillations.
The phase shift C/B moves the entire graph horizontally. A positive C shifts the graph to the right, while a negative C shifts it to the left. Phase shift is crucial in applications like AC circuits, where it describes the timing offset between voltage and current waveforms.
The D parameter moves the graph up (positive D) or down (negative D). This is the simplest transformation: it changes the midline of the function from y = 0 to y = D. In real-world modeling, this often represents a baseline offset, such as average temperature around which seasonal fluctuations occur.