Graphs of sine, cosine, and tangent
Graphs of Sine, Cosine, and Tangent
Graphs of Sine, Cosine, and Tangent is a topic in Trigonometric Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7e, which requires students to graph trigonometric functions, showing period, midline, and amplitude.
The sine and cosine graphs are smooth waves of period \(2\pi\) and amplitude 1, while the tangent graph has period \(\pi\) with vertical asymptotes.
Theory
The three basic trig graphs:
- \(y=\sin x\) — a smooth wave through the origin, period \(2\pi\), range \([-1,1]\).
- \(y=\cos x\) — the same wave shifted a quarter period; it starts at its maximum \((0,1)\).
- \(y=\tan x\) — period \(\pi\), with vertical asymptotes where \(\cos x=0\); its range is all reals.
Sine and cosine are best drawn with five key points per period (the maxima, minima, and zeros).
Basic features:
How to sketch a trig graph
- Sine/cosine: mark the midline and amplitude, then plot the five key points across one period and repeat.
- Tangent: draw the asymptotes at \(\cos x=0\), then a rising branch through the origin between each pair.
- Extend periodically in both directions.
Sine repeats every full turn and oscillates between \(-1\) and \(1\).
| \(\text{period}\) | \(=\) | \(2\pi\) |
| \(\text{range}\) | \(=\) | \([-1,1]\) |
Cosine starts at its maximum and steps through zero, min, zero, max.
| \((0,1)\) | max | |
| \((\dfrac{\pi}{2},0)\) | zero | |
| \((\pi,-1)\) | min | |
| \((\dfrac{3\pi}{2},0)\) | zero | |
| \((2\pi,1)\) | max |
Tangent is \(\dfrac{\sin x}{\cos x}\), undefined where \(\cos x=0\).
| \(\cos x=0\) | \(\Rightarrow\) | \(x=\dfrac{\pi}{2}+\pi n\) |
Asymptotes at every odd multiple of \(\dfrac{\pi}{2}\).
Sine is zero at every integer multiple of \(\pi\).
| \(\sin x=0\) | \(\Rightarrow\) | \(x=\pi n,\ n\in\mathbb{Z}\) |
Common pitfalls
Frequently asked questions
What are the period and amplitude of sine and cosine?
Both have period \(2\pi\) and amplitude 1, oscillating between \(-1\) and \(1\).
How is the cosine graph related to the sine graph?
Cosine is sine shifted left by \(\dfrac{\pi}{2}\): \(\cos x=\sin(x+\dfrac{\pi}{2})\). It starts at its maximum.
Why does the tangent graph have asymptotes?
Because \(\tan x=\dfrac{\sin x}{\cos x}\) is undefined where \(\cos x=0\), namely at \(x=\dfrac{\pi}{2}+\pi n\).
What is the period of tangent?
\(\pi\). Tangent repeats every \(\pi\), twice as often as sine and cosine.