Graphs of Sine and Cosine

Sine and cosine graphs turn circular motion into smooth repeating waves described by amplitude, period, midline, and phase.

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Amplitude, period, and phase shift describe a repeating graph. A wave repeats across the axis. A vertical measure marks amplitude, a horizontal bracket marks one period, and a second faint wave shows how a phase shift moves the cycle.

Sine and cosine trace the vertical and horizontal coordinates of uniform motion around a circle.

Parent graphs

The parent functions are

y=\sin x\qquad\text{and}\qquad y=\cos x.

Both have domain all real numbers, range $[-1,1]$, amplitude $1$, midline $y=0$, and period $360^\circ$ or $2\pi$ radians.

Sine begins at its midline and rises. Cosine begins at a maximum. Their shapes are identical except for a horizontal shift.

Transformation form

A transformed sinusoidal function can be written

y=a\sin(k(x-d))+c

y=a\cos(k(x-d))+c.

Its amplitude is $|a|$, its midline is $y=c$, and its phase shift is $d$. A negative $a$ reflects the wave across its midline.

Worked example: analyze a sine graph

Worked example: analyze a reflected cosine

For

y=-2\cos x+5,

the amplitude is $2$, midline $y=5$, period $360^\circ$, maximum $7$, and minimum $3$. Because $a<0$, the graph begins at a minimum rather than a maximum when $x=0$.

The range is $[5-2,5+2]=[3,7]$.

Finding an equation from a graph

Read maximum $M$ and minimum $m$ first:

\text{amplitude}=\frac{M-m}{2},qquad \text{midline}=\frac{M+m}{2}.

Measure the horizontal distance between repeating matching points to find the period, then calculate $k$. Choose sine when a convenient starting point is a midline crossing; choose cosine when it is a maximum or minimum. Multiple equivalent equations may model the same graph.

Common mistakes

Calling $a$ the maximum. The maximum is $c+|a|$; amplitude is $|a|$.

Using $k$ as the period. Period is $360^\circ/|k|$ or $2\pi/|k|$.

Forgetting to factor the inside. Rewrite $2x-60^\circ$ as $2(x-30^\circ)$.

Spacing points by the whole period. Key points are one quarter-period apart.

Connecting key points with straight lines. The graph changes smoothly.