The Cosine Function, foundations for calculus lesson

Watch a turning wheel from directly above and the bolt on its rim seems to slide back and forth along a straight line. That side to side position, on a wheel of radius one, is the cosine of the angle turned. Plotted against the angle it makes a wave just like the sine wave, only starting at its peak instead of at zero.

Key terms

Cosine as a horizontal coordinate

Definition For a real number $x$ , let $P(x)$ be the point where the terminal side of the angle of $x$ radians in standard position meets the unit circle. The cosine of $x$ is the horizontal coordinate of that point,

P(x) = (\cos x, \sin x).

This defines a function $\cos$ on all real numbers, with range $[-1, 1]$ . The value $1$ is reached exactly at $x = 2\pi k$ and the value $-1$ exactly at $x = \pi + 2\pi k$ , for integers $k$ .

In plain words Turn through the angle and read how far right of the center the point is. A point left of center gives a negative cosine.

Example The angle $\frac{3\pi}{4}$ lands in quadrant II with reference angle $\frac{\pi}{4}$ . The point is left of center by $\frac{\sqrt{2}}{2}$ , so

\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}.

Cosine graph

Definition The graph of cosine is the set of points $(x, \cos x)$ for all real $x$ . Here the horizontal axis of the graph carries the input $x$ , and the height of the graph is the horizontal coordinate of $P(x)$ on the circle. One cycle over $[0, 2\pi]$ has the landmarks

$x$	$0$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$
$\cos x$	$1$	$0$	$-1$	$0$	$1$

Cosine decreases on $[0, \pi]$ and increases on $[\pi, 2\pi]$ . Its graph is the sine graph shifted left by $\frac{\pi}{2}$ , since

\begin{gathered} \cos x = \sin\left(x + \frac{\pi}{2}\right) \\[0.4em] \text{for every real } x. \end{gathered}

In plain words Take the side to side position of the moving point and plot it as a height above each input. The wave starts at its top, falls through zero to its bottom at $\pi$ , and climbs back.

Example More points on the first half cycle come from the special triangles.

$x$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$\cos x$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$

The drop is slow near $0$ and $\pi$ and fastest as the graph crosses zero at $\frac{\pi}{2}$ .

Even symmetry

Definition A function $f$ is even if $-x$ is in its domain whenever $x$ is and $f(-x) = f(x)$ for all such $x$ . Cosine is even,

\cos(-x) = \cos x \quad \text{for every real } x.

The point $P(-x)$ is the reflection of $P(x)$ across the horizontal axis, and that reflection does not change the horizontal coordinate. The graph of an even function is symmetric about the vertical axis.

In plain words Turning clockwise instead of counterclockwise by the same amount lands you at a point straight below or above, with the same left or right position.

Example From $\cos \frac{2\pi}{3} = -\frac{1}{2}$ we get at once

\cos\left(-\frac{2\pi}{3}\right) = -\frac{1}{2}.

Compare sine, which is odd. Here $\sin\left(-\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}$ while $\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$ .

Period

Definition For every real $x$ and every integer $k$

\cos(x + 2\pi k) = \cos x,

because adding $k$ full turns returns the point to the same place. The least positive period of cosine is $2\pi$ . To see that nothing smaller works, suppose $p > 0$ is a period. Then $\cos p = \cos(0 + p) = \cos 0 = 1$ . Cosine equals $1$ only at the inputs $2\pi k$ , so $p = 2\pi k$ for a positive integer $k$ , and therefore $p \ge 2\pi$ .

In plain words The side to side motion repeats after one full turn and not sooner.

Example The input $-\frac{17\pi}{3}$ plus three full turns is

-\frac{17\pi}{3} + 6\pi = \frac{\pi}{3},

so $\cos\left(-\frac{17\pi}{3}\right) = \cos \frac{\pi}{3} = \frac{1}{2}$ .

Zeros

Definition The zeros of cosine are the real numbers $x$ with $\cos x = 0$ . They are exactly

x = \frac{\pi}{2} + k\pi, \qquad k \text{ an integer}.

These are the inputs whose point is $(0, 1)$ or $(0, -1)$ , the top and bottom of the circle.

In plain words The horizontal coordinate is zero only when the point is straight above or straight below the center, which happens every half turn starting at a quarter turn.

Example The zeros of cosine in $[-\pi, 2\pi]$ come from $k = -1$ , $0$ and $1$ . They are $-\frac{\pi}{2}$ , $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ .

Amplitude

Definition For real numbers $A \ne 0$ , $B \ne 0$ and $D$ , the function

y = A\cos(Bx) + D

has amplitude $\lvert A \rvert$ and least positive period $\frac{2\pi}{\lvert B \rvert}$ . Its values fill the interval $[D - \lvert A \rvert, D + \lvert A \rvert]$ . The amplitude is the distance from the midline to a peak, not the height of the peak itself.

In plain words Amplitude measures how far the wave swings away from its middle level. Lifting the whole wave does not change the swing.

Example The function $y = 3\cos x + 5$ has peaks at height $8$ and troughs at height $2$ . Its amplitude is $3$ , not $8$ , because the peaks are $3$ above the middle level $y = 5$ .

Midline

Definition The midline of $y = A\cos(Bx) + D$ is the horizontal line $y = D$ , halfway between the maximum $D + \lvert A \rvert$ and the minimum $D - \lvert A \rvert$ . From the maximum $M$ and minimum $m$ of the wave

D = \frac{M + m}{2}, \qquad \lvert A \rvert = \frac{M - m}{2}.

In plain words The midline is the level the wave oscillates around. Shifting the graph up or down by $D$ moves the midline with it.

Example Build one full cycle of $y = -2\cos\left(\frac{x}{2}\right) + 1$ . The period is

\frac{2\pi}{1/2} = 4\pi,

the amplitude is $2$ and the midline is $y = 1$ . Each landmark $(x, y)$ of $\cos x$ moves to $(2x, -2y + 1)$ , where doubling the input stretches the period, the factor $-2$ stretches and reflects, and the $+1$ lifts.

$x$	$0$	$\pi$	$2\pi$	$3\pi$	$4\pi$
$y$	$-1$	$1$	$3$	$1$	$-1$

Because of the reflection the cycle starts at its minimum $-1$ , crosses the midline at $\pi$ , peaks at $3$ when $x = 2\pi$ , and returns.

Check yourself

Find $\cos \frac{5\pi}{6}$ and $\cos\left(-\frac{\pi}{4}\right)$ exactly.

Show the answer

The angle $\frac{5\pi}{6}$ is in quadrant II with reference angle $\frac{\pi}{6}$ , so $\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$ . Because cosine is even, $\cos\left(-\frac{\pi}{4}\right) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ .

A wave of the form $y = A\cos(Bx) + D$ with $A > 0$ and $B > 0$ has maximum $4$ , minimum $-2$ and period $\pi$ . Find $A$ , $B$ and $D$ .

Show the answer

The midline is $D = \frac{4 + (-2)}{2} = 1$ and the amplitude is $A = \frac{4 - (-2)}{2} = 3$ . The period gives $\frac{2\pi}{B} = \pi$ , so $B = 2$ . The wave is $y = 3\cos(2x) + 1$ .

A classmate says the amplitude of $y = \cos x - 3$ is $3$ . What is wrong?

Show the answer

The number $-3$ is the vertical shift, so the midline is $y = -3$ . The wave swings between $-4$ and $-2$ , which is $1$ unit from the midline each way, so the amplitude is $1$ .

The Cosine Function

Key terms

Cosine as a horizontal coordinate

Cosine graph

Even symmetry

Period

Zeros

Amplitude

Midline

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus

Resources for this lesson

Unit circle explorer