GTF - Graphing Sine and Cosine Lesson

Graphing Sine and Cosine

To the right, you can see a completed Unit Circle. As a reminder, the Unit Circle provides a quick reference for the trigonometric ratios of special angles.

We know that sine and cosine represent specific RATIOS, where r equals the radius:

Sine: the ratio of y to r, $\sin\theta=\frac{y}{r}$

Cosine: the ratio of x to r, $\cos\theta=\frac{y}{r}$

And, when we are working with the Unit Circle, r = 1, so the ratios simplify to $\cos\theta=x\:and\:sin\theta=y$.

Well, what if we wanted to graph the function $f\left(t\right)=\sin\left(t\right)$? Let's make a table:

t	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$f\left(t\right)=\sin t$	0	$\frac{1}{2}=0.5$	$\frac{\sqrt[]{2}}{2}\approx.707$	$\frac{\sqrt[]{3}}{2}\approx0.866$	1	$\frac{\sqrt[]{3}}{2}\approx0.866$	$\frac{\sqrt[]{2}}{2}\approx0.707$	$\frac{1}{2}\approx0.5$	0

 

t	$\frac{7\pi}{6}$	$\frac{5\pi}{4}$	$\frac{4\pi}{3}$	$\frac{3\pi}{2}$	$\frac{5\pi}{3}$	$\frac{7\pi}{4}$	$\frac{11\pi}{6}$	$2\pi$
$f\left(t\right)=\sin t$	$-\frac{1}{2}=-0.5$	$-\frac{\sqrt[]{2}}{2}\approx-0.707$	$-\frac{\sqrt[]{3}}{2}\approx-0.866$	-1	$-\frac{\sqrt[]{3}}{2}\approx-0.866$	$-\frac{\sqrt[]{2}}{2}\approx-0.707$	$-\frac{1}{2}=-0.5$	0

Now, let's plot these points:

And, if we finished the graph of $f\left(t\right)=\sin\left(t\right)$, our graph would look like this:

Let's answer a few questions about the graph of $f\left(t\right)=\sin\left(t\right)$.

1. What is the domain of $f\left(t\right)=\sin\left(t\right)$?

• Solution: all real numbers

2. What is the range of $f\left(t\right)=\sin\left(t\right)$?

• Solution: [-1, 1]

3. Is $f\left(t\right)=\sin\left(t\right)$ even, odd, or neither?

• Solution: odd

Watch this video to discuss the symmetry of $f\left(t\right)=\sin\left(t\right)$.

Now, what if we wanted to graph the function,$f\left(t\right)=\cos\left(t\right)$? Let's make a table:

t	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$f\left(t\right)=\cos t$	1	$\frac{\sqrt[]{3}}{2}\approx0.866$	$\frac{\sqrt[]{2}}{2}\approx0.707$	$\frac{1}{2}=0.5$	0	$-\frac{1}{2}=-0.5$	$-\frac{\sqrt[]{2}}{2}\approx-0.707$	$-\frac{\sqrt[]{3}}{2}\approx-0.866$	-1

 

t	$\frac{7\pi}{6}$	$\frac{5\pi}{4}$	$\frac{4\pi}{3}$	$\frac{3\pi}{2}$	$\frac{5\pi}{3}$	$\frac{7\pi}{4}$	$\frac{11\pi}{6}$	$2\pi$
$f\left(t\right)=\cos t$	$-\frac{\sqrt[]{3}}{2}\approx-0.866$	$-\frac{\sqrt[]{2}}{2}\approx-0.707$	$-\frac{1}{2}=-0.5$	0	$\frac{1}{2}=0.5$	$\frac{\sqrt[]{2}}{2}\approx0.707$	$\frac{\sqrt[]{3}}{2}\approx0.866$	1

Now, let's plot these points:

And, if we finished the graph of $f\left(t\right)=\cos\left(t\right)$, our graph would look like this:

Let's answer a few questions about the graph of $f\left(t\right)=\cos\left(t\right)$.

1. What is the domain of $f\left(t\right)=\cos\left(t\right)$?

• Solution: all real numbers

2. What is the range of $f\left(t\right)=\cos\left(t\right)$?

• Solution: [-1, 1]

3. Is $f\left(t\right)=\cos\left(t\right)$ even, odd, or neither?

• Solution: even

Watch this video to discuss the symmetry of $f\left(t\right)=\cos\left(t\right)$.

When graphing $f\left(t\right)=\sin\left(t\right)$ and $f\left(t\right)=\cos\left(t\right)$ it is not necessary to graph all of the points we did in the images above. However, we'd like to focus on 5 critical points shown below:

 IMAGES CREATED BY GAVS