Topic 3.4 Notes – Sine and Cosine Function Graphs

What Sine and Cosine Are

On the unit circle, an angle $\theta$ in standard position lands at a point
$$P=(\cos\theta,\sin\theta).$$

That single fact drives the whole topic.

In the unit-circle diagram, the point in Quadrant I shows the basic idea. The horizontal coordinate is cosine, and the vertical coordinate is sine.

• $\sin\theta$ is the signed $y$-coordinate of $P$
• $\cos\theta$ is the signed $x$-coordinate of $P$

Sine and cosine on the unit circle

“Signed” matters.

• Below the $x$-axis, sine is negative.
• Left of the $y$-axis, cosine is negative.

The graph connection is the part students mix up most:

• Unit-circle point $(\cos\theta,\sin\theta)$
• Sine graph point $(\theta,\sin\theta)$
• Cosine graph point $(\theta,\cos\theta)$

So the graph’s horizontal axis is the angle input $\theta$, not the unit-circle $x$-coordinate.

A few facts you need right away:

• Domain of both sine and cosine is all real numbers
• Range of both is $[-1,1]$
• Positive angles go counterclockwise, negative angles go clockwise
• Angles bigger than $2\pi$ mean extra revolutions

Building the Parent Graphs from the Unit Circle

The quarter-turn angles give the landmarks for one full revolution:

$$0,\quad \frac{\pi}{2},\quad \pi,\quad \frac{3\pi}{2},\quad 2\pi$$

Sine graph landmarks

At those angles, the unit-circle points are
$(1,0),(0,1),(-1,0),(0,-1),(1,0)$.

Reading the $y$-values gives the sine graph points:

• $(0,0)$
• $\left(\frac{\pi}{2},1\right)$
• $(\pi,0)$
• $\left(\frac{3\pi}{2},-1\right)$
• $(2\pi,0)$

Sine goes up to $1$, down through $0$ to $-1$, then back up to $0$.

Cosine graph landmarks

Reading the $x$-values instead gives:

• $(0,1)$
• $\left(\frac{\pi}{2},0\right)$
• $(\pi,-1)$
• $\left(\frac{3\pi}{2},0\right)$
• $(2\pi,1)$

Cosine goes down from $1$ to $-1$, then back up to $1$.

The graphs below show those five landmark points for one full cycle of each function.

Parent graphs of sine and cosine on $0$ to $2\pi$

These are smooth curves, not broken line segments. And the fastest way to tell them apart is at $\theta=0$:

• $\sin 0=0$
• $\cos 0=1$

Reading Key Features from the Circle and the Graph

The landmarks repeat every full revolution, so

$$\sin(\theta+2\pi k)=\sin\theta \qquad \cos(\theta+2\pi k)=\cos\theta$$

where $k$ is any integer.

For sine:

• zeros when $\theta=k\pi$
• maximum $1$ when $\theta=\frac{\pi}{2}+2\pi k$
• minimum $-1$ when $\theta=\frac{3\pi}{2}+2\pi k$

For cosine:

• zeros when $\theta=\frac{\pi}{2}+k\pi$
• maximum $1$ when $\theta=2\pi k$
• minimum $-1$ when $\theta=\pi+2\pi k$

One common wording trap: $1$ is the maximum output value. $\frac{\pi}{2}$ is an input where sine reaches that maximum.

Moving Between Representations

You should be able to move back and forth between circle, graph, and function values.

• From a unit-circle point, read off sine from $y$ and cosine from $x$.
• From a graph point like $(\theta,\sin\theta)$, the first coordinate is the angle and the second is the output.
• “Vertical position” means sine. “Horizontal position” means cosine.

Examples:

• If $\theta=\frac{5\pi}{6}$ and the terminal point is $\left(-\frac{\sqrt3}{2},\frac12\right)$, then
  $\sin\left(\frac{5\pi}{6}\right)=\frac12$ and the sine graph has point $\left(\frac{5\pi}{6},\frac12\right)$
  $\cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt3}{2}$ and the cosine graph has point $\left(\frac{5\pi}{6},-\frac{\sqrt3}{2}\right)$
• If $\theta=-\frac{\pi}{2}$, the terminal point is $(0,-1)$, so
  $\sin\left(-\frac{\pi}{2}\right)=-1$ and $\cos\left(-\frac{\pi}{2}\right)=0$

A single sine or cosine value usually matches more than one angle, so the output alone usually does not give one unique input.

Common Mix-Ups

• The graph’s horizontal axis is $\theta$, not the unit-circle $x$-coordinate.
• Sine and cosine are signed coordinates, not plain distances.
• Sine = vertical, cosine = horizontal.
• Both parent graphs stay between $-1$ and $1$.
• The curve should look wavelike and smooth.
• Cosine “increasing” means the $x$-coordinate is increasing. The distance from the origin never changes on the unit circle.