Topic 3.3 Notes – Sine and Cosine Function Values

Sine and Cosine as Circle Coordinates

An angle is in standard position when its vertex is at the origin and its initial side lies on the positive $x$-axis. The terminal ray lands on a circle of radius $r$.

That intersection point is

$$P=(r\cos\theta,\; r\sin\theta)$$

On the unit circle, where $r=1$, this becomes

$$P=(\cos\theta,\sin\theta)$$

So the whole idea is simple:

• cosine = $x$-coordinate
• sine = $y$-coordinate

Unit circle coordinates

In the diagram, the point on the unit circle has coordinates $(x,y)$, so $x=\cos t$ and $y=\sin t$.

You also use this backward. If $P=(x,y)$ is on a circle of radius $r$, then

$$\cos\theta=\frac{x}{r} \qquad \sin\theta=\frac{y}{r}$$

The radius is a scale factor. It multiplies both coordinates, not the angle.

Quick checks that catch lots of errors:

• signs must match the quadrant
• each coordinate must stay between $-r$ and $r$
• the point must satisfy $x^2+y^2=r^2$

Exact Values You Need to Know

Some angles come straight from the axes.

Axis angles

On the unit circle:

• $0$ and $2\pi$ give $(1,0)$
• $\frac{\pi}{2}$ gives $(0,1)$
• $\pi$ gives $(-1,0)$
• $\frac{3\pi}{2}$ gives $(0,-1)$

On radius $r$, just scale those to $(r,0)$, $(0,r)$, $(-r,0)$, $(0,-r)$.

Special-angle families

These come from special right triangles.

This unit circle collects the exact coordinates for the axis angles and the most common special angles in both degrees and radians.

Common exact values on the unit circle

Keep exact values in radical form, not decimals, unless the question asks for an approximation.

Quadrants, Reference Angles, and Coterminal Angles

A reference angle is the acute angle the terminal ray makes with the $x$-axis. It gives the coordinate magnitudes. The quadrant gives the signs.

• Quadrant I: $(+,+)$
• Quadrant II: $(-,+)$
• Quadrant III: $(-,-)$
• Quadrant IV: $(+,-)$

Since cosine is $x$ and sine is $y$, those sign patterns follow automatically.

Coterminal angles differ by $2\pi k$, where $k$ is any integer. They land on the same terminal ray, so they have the same sine and cosine.

Negative angles often get easier if you rewrite them in $[0,2\pi)$. For example, $-\frac{3\pi}{4}$ is coterminal with $\frac{5\pi}{4}$.

One easy trap: some multiples of $\frac{\pi}{6}$ are actually axis angles, like $\frac{\pi}{2}$. Read those directly from the axes.

Finding a Point or a Trig Value from an Angle

Here’s the full path the test expects you to be able to do cleanly:

1. Decide the case: axis angle, $\pi/4$ family, $\pi/6$ family, or $\pi/3$ family.
2. If needed, rewrite to a coterminal angle in one revolution.
3. Find the quadrant or see if it lies on an axis.
4. If it is in a quadrant, find the reference angle.
5. Use the right special-angle magnitudes.
6. Apply signs from the quadrant.
7. If the radius is not 1, multiply both coordinates by $r$.

Example. $\frac{5\pi}{6}$ on radius $8$:

• reference angle $= \frac{\pi}{6}$
• QII gives signs $(-,+)$
• unit-circle point $= \left(-\frac{\sqrt3}{2}, \frac12\right)$
• radius 8 gives $(-4\sqrt3, 4)$

If you’re given a point, work backward.

Example. $P=(-5,12)$:

• $r=\sqrt{(-5)^2+12^2}=13$
• $\cos\theta=\frac{-5}{13}$
• $\sin\theta=\frac{12}{13}$

If $\sin\theta=-\frac{\sqrt3}{2}$ and $0\le \theta<2\pi$, then $y=-\frac{\sqrt3}{2}$, so the angles are $\frac{4\pi}{3}$ and $\frac{5\pi}{3}$.

Common Mistakes and Fast Self-Checks