Trigonometry Reference
Unit Circle
On the unit circle, a point has coordinates $(\cos\theta,\sin\theta)$. The goal here is not brute memorization of every entry, but remembering a few patterns that let you rebuild the circle quickly under pressure.
Solved Examples
Jump directly to a worked example by angle or quadrant pattern.
Fact Table
| Pattern | What to Remember | Why It Helps |
|---|---|---|
| Bowtie | The numerators $\sqrt1,\sqrt2,\sqrt3$ swap places between cosine and sine. | You can rebuild the first-quadrant coordinates without memorizing two separate lists. |
| $6,4,3$ | The denominator pattern in radians is $\pi/6,\pi/4,\pi/3$. | This organizes the three interior first-quadrant angles instantly. |
| Reference angles repeat | $30^\circ,\ 45^\circ,\ 60^\circ$ reappear in every quadrant. | Only the signs and full-angle formulas change. |
| Quadrant signs | Use the quadrant to decide the signs of cosine, sine, and tangent. | The absolute values stay tied to the same reference-angle pattern. |
Content Formulas
Unit Circle Point
$$(x,y)=(\cos\theta,\sin\theta)$$
Tangent
$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac yx,\quad x\ne0$$
Bowtie Pattern
$$\cos\theta=\frac{\sqrt3}{2},\frac{\sqrt2}{2},\frac12$$ $$\sin\theta=\frac12,\frac{\sqrt2}{2},\frac{\sqrt3}{2}$$
Radians Pattern
$$30^\circ,45^\circ,60^\circ\longleftrightarrow \frac\pi6,\frac\pi4,\frac\pi3$$
Build everything from Quadrant I first. Then copy the same absolute values into the other quadrants and change only the signs.
Bowtie and 6-4-3 Patterns
| Angle | Radians Pattern | Cosine | Sine | Memory Cue |
|---|---|---|---|---|
| $30^\circ$ | $\pi/6$ | $\sqrt3/2$ | $1/2$ | Starts with denominator $6$, and sine gets the smaller numerator. |
| $45^\circ$ | $\pi/4$ | $\sqrt2/2$ | $\sqrt2/2$ | Middle angle, perfectly balanced. |
| $60^\circ$ | $\pi/3$ | $1/2$ | $\sqrt3/2$ | Ends with denominator $3$, and sine gets the larger numerator. |
Many students remember this as a bowtie: cosine goes $\sqrt3,\sqrt2,1$ while sine goes $1,\sqrt2,\sqrt3$, all over $2$.
Quadrant II Shortcut
| Reference Angle | Quadrant I | Quadrant II | Pattern |
|---|---|---|---|
| 30 degrees | pi/6 | 5pi/6 | Numerator is denominator minus 1: 5 vs. 6. |
| 45 degrees | pi/4 | 3pi/4 | Numerator is denominator minus 1: 3 vs. 4. |
| 60 degrees | pi/3 | 2pi/3 | Numerator is denominator minus 1: 2 vs. 3. |
Quadrant II reverses the first-quadrant denominator sequence: 6, 4, 3 becomes 5, 3, 2.
Quadrant III Shortcut
| Reference Angle | Quadrant I | Quadrant III | Pattern |
|---|---|---|---|
| $30^\circ$ | $\pi/6$ | $7\pi/6$ | Numerator is 1 more than the denominator: $7$ vs. $6$. |
| $45^\circ$ | $\pi/4$ | $5\pi/4$ | Numerator is 1 more than the denominator: $5$ vs. $4$. |
| $60^\circ$ | $\pi/3$ | $4\pi/3$ | Numerator is 1 more than the denominator: $4$ vs. $3$. |
That “one more in the numerator” pattern is a quick way to spot the three standard Quadrant III angles in radians.
Quadrant IV Shortcut
| Reference Angle | Quadrant I | Quadrant IV | Pattern |
|---|---|---|---|
| 30 degrees | pi/6 | 11pi/6 | Numerator is two times the denominator minus 1: 11 = 2(6) - 1. |
| 45 degrees | pi/4 | 7pi/4 | Numerator is two times the denominator minus 1: 7 = 2(4) - 1. |
| 60 degrees | pi/3 | 5pi/3 | Numerator is two times the denominator minus 1: 5 = 2(3) - 1. |
Q4 is the “almost 2” pattern: 11/6, 7/4, 5/3. Start at 2pi and subtract the reference angle.
Tangent Pattern Table
| Reference Angle | Tangent | Why |
|---|---|---|
| $30^\circ$ | $\sqrt3/3$ | $\frac{1/2}{\sqrt3/2}=\frac1{\sqrt3}=\sqrt3/3$ |
| $45^\circ$ | $1$ | $\frac{\sqrt2/2}{\sqrt2/2}=1$ |
| $60^\circ$ | $\sqrt3$ | $\frac{\sqrt3/2}{1/2}=\sqrt3$ |
Tangent has a clean rise: $\sqrt3/3,\ 1,\ \sqrt3$. Then just apply the quadrant sign rule.
Six-Function Memory Table
| Angle | Sine | Cosine | Tangent | Cosecant | Secant | Cotangent |
|---|---|---|---|---|---|---|
| $0$ | $0$ | $1$ | $0$ | undefined | $1$ | undefined |
| $\pi/6$ | $1/2$ | $\sqrt3/2$ | $\sqrt3/3$ | $2$ | $2\sqrt3/3$ | $\sqrt3$ |
| $\pi/4$ | $\sqrt2/2$ | $\sqrt2/2$ | $1$ | $\sqrt2$ | $\sqrt2$ | $1$ |
| $\pi/3$ | $\sqrt3/2$ | $1/2$ | $\sqrt3$ | $2\sqrt3/3$ | $2$ | $\sqrt3/3$ |
| $\pi/2$ | $1$ | $0$ | undefined | $1$ | undefined | $0$ |
Mnemonic: build sine and cosine from the bowtie, divide to get tangent, then flip sine, cosine, and tangent to get cosecant, secant, and cotangent.
Classic Examples
Coordinates from an Angle
Find the unit-circle point for $150^\circ$.
Solution Steps
- Write $150^\circ$ as $180^\circ-30^\circ$ to identify its reference angle and quadrant.
- Use the Quadrant II sign for cosine.
- Use the first-quadrant sine value for the $30^\circ$ reference angle.
- State the unit-circle point.
$$150^\circ=180^\circ-30^\circ$$
$$\cos150^\circ=-\frac{\sqrt3}{2}$$
$$\sin150^\circ=\frac12$$
$$\left(-\frac{\sqrt3}{2},\frac12\right)$$
Find Tangent
Find $\tan(225^\circ)$.
Solution Steps
- Write $225^\circ$ as $180^\circ+45^\circ$ to identify its reference angle and quadrant.
- Use the Quadrant III sign for tangent and state $\tan(225^\circ)=1$.
$$225^\circ=180^\circ+45^\circ$$
$$\tan(225^\circ)=1$$
Use the Quadrant III Radian Pattern
Find the angle in Quadrant III with reference angle $60^\circ$, then give its unit-circle point.
Solution Steps
- Convert the $60^\circ$ reference angle to $\pi/3$.
- Use the Quadrant III pattern to find the full angle.
- Use the Quadrant III cosine sign.
- Use the Quadrant III sine sign.
- State the unit-circle point.
$$60^\circ\longleftrightarrow \frac\pi3$$
$$\text{Quadrant III angle }=\frac{4\pi}{3}$$
$$\cos\left(\frac{4\pi}{3}\right)=-\frac12$$
$$\sin\left(\frac{4\pi}{3}\right)=-\frac{\sqrt3}{2}$$
$$\left(-\frac12,-\frac{\sqrt3}{2}\right)$$