Graphs of Sine and Cosine
The sine and cosine functions are the foundation of trigonometry and appear throughout mathematics, physics, engineering, and music. Unlike algebraic functions whose graphs are lines or parabolas, sine and cosine produce smooth, repeating wave shapes called sinusoids. Understanding how to construct and read these graphs is one of the most powerful tools you will develop in high school mathematics. Both y = sin(x) and y = cos(x) are defined for every real number x, where x is measured in radians. Their outputs are always between −1 and 1, inclusive. The graphs repeat their pattern every 2π radians — a property called periodicity. This means that once you understand one full cycle, you understand the entire infinite graph. In this article you will learn to plot the key points of each graph from scratch, identify the amplitude, period, and phase, and interpret transformations such as vertical stretches and horizontal shifts. These skills connect directly to real-world wave phenomena including sound, light, and alternating current electricity.
- Understand the Unit Circle Connection
- Plot the Five Key Points of sin(x)
- Plot the Five Key Points of cos(x)
- Identify Amplitude and Period
- Apply Transformations
Understand the Unit Circle Connection
For any angle x (in radians), sin(x) is the y-coordinate and cos(x) is the x-coordinate of the point on the unit circle at angle x. As x increases from 0 to 2π, you travel once around the circle. The height (y-coordinate) traces the sine wave; the horizontal position (x-coordinate) traces the cosine wave. This geometric origin explains every feature of the graphs.
Plot the Five Key Points of sin(x)
One full cycle of y = sin(x) on [0, 2π] is captured by five landmark points: (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0). These correspond to the start, maximum, midline crossing, minimum, and return. Plot these five points on a coordinate grid, then connect them with a smooth S-shaped curve. The curve rises from 0, peaks at 1, descends back through 0, dips to −1, and returns to 0.
Plot the Five Key Points of cos(x)
One full cycle of y = cos(x) on [0, 2π] uses the landmark points: (0, 1), (π/2, 0), (π, −1), (3π/2, 0), (2π, 1). The cosine graph starts at its maximum, descends to 0, drops to its minimum at π, rises back to 0, and returns to 1. Notice that the cosine graph is simply the sine graph shifted π/2 units to the left: cos(x) = sin(x + π/2).
Identify Amplitude and Period
The amplitude of y = sin(x) and y = cos(x) is 1 — the maximum distance from the midline (y = 0) to a peak or trough. The period is 2π ≈ 6.283 radians, the horizontal length of one complete cycle. For the general forms y = A sin(Bx) and y = A cos(Bx), the amplitude is |A| and the period is 2π/|B|. For example, y = 3 sin(2x) has amplitude 3 and period π.
Apply Transformations
The full transformation form is y = A sin(Bx − C) + D. Here |A| is amplitude, 2π/|B| is the period, C/B is the phase shift (horizontal translation), and D is the vertical shift (midline). A negative A reflects the graph over the x-axis. For example, y = 2 cos(x − π/3) + 1 has amplitude 2, period 2π, phase shift π/3 to the right, and midline y = 1. Always identify these four parameters before sketching.
The Key Equation
- A — Amplitude (dimensionless): Half the total height of the wave; the maximum value is D + |A| and the minimum is D − |A|.
- B — Angular frequency (radians⁻¹): Controls how compressed or stretched the wave is horizontally. The period equals 2π divided by |B|.
- C — Phase constant (radians): Used to calculate the phase shift C/B, which moves the graph left (positive shift when C > 0) or right.
- D — Vertical shift (dimensionless): The value of the midline y = D; shifts the entire wave up (D > 0) or down (D < 0).
Worked Example
Problem. Sketch one full period of y = 2 sin(πx − π) + 1. State the amplitude, period, phase shift, and midline, then find the five key x-values and their y-values.
Answer. Amplitude = 2, Period = 2, Phase shift = 1 unit right, Midline: y = 1. Key points: (1, 1), (1.5, 3), (2, 1), (2.5, −1), (3, 1).
Origins · 1 minLeonhard Euler and how this was discovered
Euler established the modern notation for sine and cosine functions and proved the landmark identity e^(ix) = cos(x) + i·sin(x), now called Euler's formula. This equation unifies trigonometry, exponential functions, and complex numbers, and it is the reason that sinusoidal functions appear throughout modern physics and engineering. His 1748 work Introductio in analysin infinitorum treated trigonometric functions as ratios of real numbers rather than merely as geometric lengths, making the graphical approach we use today possible. Euler lost sight in his right eye in 1738 and became almost completely blind by 1771, yet he produced roughly half of his total mathematical output after losing his vision — dictating his work to assistants.
Related Concepts
- Transformations of Sinusoidal Functions — Once you can graph basic sine and cosine, the next step is applying amplitude changes, period changes, phase shifts, and vertical shifts to produce any sinusoidal function.
- The Unit Circle — Sine and cosine graph values come directly from the coordinates of points on the unit circle; understanding the circle gives geometric meaning to every feature of the wave.
- Graph of Tangent and Cotangent — Tangent is defined as sin(x)/cos(x), so its graph's asymptotes and zeros follow directly from patterns seen in the sine and cosine graphs.
- Pythagorean Trigonometric Identities — The identity sin²(x) + cos²(x) = 1 is visible in the graphs: wherever one function has value 1, the other is 0, and their squared values always sum to 1.
- Periodic Functions — Sine and cosine are the most important examples of periodic functions; studying periodicity in general deepens your understanding of why the graphs repeat every 2π.