How to graph sine cosine and tangent?

Plotting sine and cosine step by step

To understand these trigonometric curves, start with the unit circle. Both sine and cosine have a period of 2π radians (360 degrees) and bounce between y = -1 and y = 1. This means their range is [-1, 1] and their domain is all real numbers.

Divide one period into four equal quarters: 0, π/2, π, 3π/2, and 2π. For y = sin(x), mark the five anchor points: (0, 0), (π/2, 1), (π, 0), (3π/2, -1), and (2π, 0). Connect them with a smooth, continuous wave.

For y = cos(x), use the same x-values but shifted: (0, 1), (π/2, 0), (π, -1), (3π/2, 0), and (2π, 1). The wave shape is identical to sine, just shifted left by π/2 radians.

Graphing tangent and finding asymptotes

Tangent is defined as sine divided by cosine. Because you cannot divide by zero, tan(x) is undefined whenever cos(x) = 0. This creates dashed vertical asymptotes at x = π/2, 3π/2, and every odd multiple of π/2.

The period of tangent is π, not 2π. Between -π/2 and π/2, tangent passes through the origin (0, 0). It crosses (-π/4, -1) and (π/4, 1), curving upward toward positive infinity on the right and downward toward negative infinity on the left. Its range is all real numbers.

Common graphing mistakes to avoid

A frequent error is drawing sharp, pointed peaks instead of smooth, rounded crests on sine and cosine waves. Keep the curves gentle like gentle water ripples.

Another common mistake is forgetting that tangent repeats twice as fast as sine and cosine. Make sure your tangent branches repeat every π units along the x-axis, never crossing their vertical asymptotes.