$y=\cos x$ Cosine starts at $\cos0=1$, with quarter-cycle values $1,0,-1,0,1$. Function$0^\circ$ ($0$)$90^\circ$ ($\frac{\pi}{2}$)$180^\circ$ ($\pi$)$270^\circ$ ($\frac{3\pi}{2}$)$360^\circ$ ($2\pi$) $\cos x$ $y=a\cos(bx)$, $a,b>0$ $\text{Amplitude}=a\qquad T=\dfrac{360^\circ}{b}\text{ or}\hspace{0.5em}\dfrac{2\pi}{b}$ FunctionAmplitudePeriod (degrees)Period (radians)Maximum value of $y$Minimum value of $y$ $y=10\cos 3x$ $y=4\cos 2x$ $y=\cos\frac{2x}{3}$ $y=\cos\frac{\pi x}{2}$ Mark quarter-period steps and the pattern $a,0,-a,0,a$. Label the extremes and stop at the stated endpoints. Example: $y=5\cos2\pi x$, $0\le x\le3$ Amplitude $5$; period $1$. Quarter-period points: $(0,5)$, $\left(\dfrac14,0\right)$, $\left(\dfrac12,-5\right)$, $\left(\dfrac34,0\right)$, $(1,5)$. Repeat for three cycles.

Sketching of Cosine Curve

$y=\cos x$

Cosine starts at $\cos0=1$, with quarter-cycle values $1,0,-1,0,1$.

Function	$0^\circ$ ($0$)	$90^\circ$ ($\frac{\pi}{2}$)	$180^\circ$ ($\pi$)	$270^\circ$ ($\frac{3\pi}{2}$)	$360^\circ$ ($2\pi$)
$\cos x$

$y=a\cos(bx)$, $a,b>0$

$\text{Amplitude}=a\qquad T=\dfrac{360^\circ}{b}\text{ or}\hspace{0.5em}\dfrac{2\pi}{b}$

Function	Amplitude	Period (degrees)	Period (radians)	Maximum value of $y$	Minimum value of $y$
$y=10\cos 3x$
$y=4\cos 2x$
$y=\cos\frac{2x}{3}$
$y=\cos\frac{\pi x}{2}$

Mark quarter-period steps and the pattern $a,0,-a,0,a$. Label the extremes and stop at the stated endpoints.

Example: $y=5\cos2\pi x$, $0\le x\le3$

Amplitude $5$; period $1$. Quarter-period points: $(0,5)$, $\left(\dfrac14,0\right)$, $\left(\dfrac12,-5\right)$, $\left(\dfrac34,0\right)$, $(1,5)$. Repeat for three cycles.

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Function	Amplitude	Period (degrees)	Period (radians)	Maximum value of \(y\)	Minimum value of \(y\)
\(y=10\cos 3x\)
\(y=4\cos 2x\)
\(y=\cos\frac{2x}{3}\)
\(y=\cos\frac{\pi x}{2}\)