The Unit Circle

Trigonometry connects the angles of a triangle to the lengths of its sides. Three ratios named sine, cosine, and tangent do most of the work.

The Unit Circle: the central idea

On the unit circle, the point reached by angle $\theta$ is $(\cos\theta,\sin\theta)$. One diagram therefore stores exact trig values and their signs.

Words you need

Unit circle

The unit circle is the circle of radius exactly one unit centered at the origin, and its radius of $1$ is what makes the coordinates of a point equal to the cosine and sine of its angle.

Standard position

An angle is in standard position when its vertex sits at the origin and its starting arm lies along the positive $x$-axis, which is the setup every unit-circle value assumes.

Terminal side

The terminal side is the arm of the angle after it has been swept, and the spot where it crosses the unit circle is the point $(\cos\theta,\sin\theta)$.

Reference angle

A reference angle is the sharp, positive angle between the terminal side and the $x$-axis, and it is always between $0$ and $90^\circ$, so it lets you reuse the five first-quadrant values everywhere.

Radian

A radian is the angle that wraps one radius length of arc around the circle, so a full turn is $2\pi$ radians and $\pi$ radians equals $180^\circ$.

Quadrant

A quadrant is one of the four regions the axes cut the plane into, numbered I to IV counterclockwise, and the quadrant a terminal side lands in decides the signs of the trig values.

What to know before this lesson

Know radians, coordinate quadrants, right-triangle ratios, and the Pythagorean theorem.

If one of those prerequisites is uncertain, use the Trigonometry subject guide to locate the earlier concept before memorizing a procedure.

The unit circle: a worked example

Every step, with the arithmetic

1. Step 1 - Convert to degrees to get your bearings$\dfrac{4\pi}{3}\times\dfrac{180}{\pi}=240^\circ$
2. Step 2 - Find the quadrant$240^\circ$ falls between $180^\circ$ and $270^\circ$, so the terminal side lands in quadrant III.
3. Step 3 - Find the reference angle$240^\circ-180^\circ=60^\circ$, which is $\dfrac{4\pi}{3}-\pi=\dfrac{\pi}{3}$.
4. Step 4 - Look up the point for that reference angleAt $60^\circ$ the first-quadrant point is $\left(\tfrac12,\tfrac{\sqrt3}{2}\right)$.
5. Step 5 - Attach the quadrant III signsIn quadrant III both coordinates are negative, so the point is $\left(-\tfrac12,-\tfrac{\sqrt3}{2}\right)$.
6. Step 6 - Read cosine and sine off the point$\cos\dfrac{4\pi}{3}=-\dfrac12$ and $\sin\dfrac{4\pi}{3}=-\dfrac{\sqrt3}{2}$
7. Step 7 - Divide for tangent$\tan\dfrac{4\pi}{3}=\dfrac{-\sqrt3/2}{-1/2}=\sqrt3$. Two negatives divide to a positive, which matches tangent being positive in quadrant III.
8. Step 8 - Check the point really is on the circle$\left(-\tfrac12\right)^2+\left(-\tfrac{\sqrt3}{2}\right)^2=\tfrac14+\tfrac34=1$

At $\pi/3$, the first-quadrant reference triangle gives coordinates $(1/2,\sqrt3/2)$.

Exact first-quadrant values

The step-by-step method for the unit circle

1. Convert the angle to a reference angle between $0$ and $\pi/2$.
2. Use the special-triangle coordinates for that reference angle.
3. Apply signs from the terminal quadrant and read cosine from $x$, sine from $y$, and tangent from $y/x$.

How to check your answer

Check $(1/2)^2+(\sqrt3/2)^2=1$ and confirm both coordinates are positive in quadrant I.

Confirm the quadrant and sign, convert degrees and radians consistently, and substitute solutions into the original interval because periodic equations usually have more than one answer.

A mistake that changes the mathematics

The ordered pair is $(\cos\theta,\sin\theta)$, not the reverse. Cosine is horizontal and sine is vertical.

Where you will actually use this

Alternating current

A spinning generator produces voltage that traces a sine wave, because the coil's position on a circle is what the wire feels. Mains power at $60$ Hz means the rotor sweeps $2\pi$ radians sixty times a second, so the voltage is $v=170\sin(120\pi t)$.

Ferris wheels and circular motion

A wheel of radius $25$ metres with its hub $27$ metres up puts a rider at height $27+25\sin\theta$. At $\theta=\pi/6$ that is $27+25(0.5)=39.5$ metres. Any rotating part, from a fan blade to a satellite, is tracked the same way.

Computer graphics

To place an object at distance $r$ and angle $\theta$ from a pivot, a program computes $(r\cos\theta,\,r\sin\theta)$. That single line puts enemies in a circle around a player, spaces spokes on a wheel, and aims a turret.

How the unit circle connects to the rest of trigonometry

• Pythagorean theorem — Every unit-circle point sits on a right triangle with legs $\cos\theta$ and $\sin\theta$ and hypotenuse $1$, so the theorem gives $\cos^2\theta+\sin^2\theta=1$ for free.
• Trigonometric identities — The identities are the unit circle written as algebra, so once the circle is memorized the Pythagorean and quotient identities stop being facts to memorize and start being things to read.
• Equation of a circle — The unit circle is the case $h=0$, $k=0$, and $r=1$ of the center-radius form, which is where $x^2+y^2=1$ comes from.

Try a transfer problem

Find all six trigonometric values at $5\pi/6$ and explain the signs using quadrant II.

Show the worked answer

The angle $5\pi/6$ is $150^\circ$, which lands in quadrant II. Its reference angle is $\pi-5\pi/6=\pi/6$, or $30^\circ$, whose first-quadrant point is $\left(\tfrac{\sqrt3}{2},\tfrac12\right)$. In quadrant II the across value is negative and the up value is positive, so the point is $\left(-\tfrac{\sqrt3}{2},\tfrac12\right)$. That gives $\cos\tfrac{5\pi}{6}=-\tfrac{\sqrt3}{2}$ and $\sin\tfrac{5\pi}{6}=\tfrac12$. Dividing, $\tan\tfrac{5\pi}{6}=\tfrac{1/2}{-\sqrt3/2}=-\tfrac{1}{\sqrt3}=-\tfrac{\sqrt3}{3}$. The reciprocals follow: $\csc=2$, $\sec=-\tfrac{2}{\sqrt3}=-\tfrac{2\sqrt3}{3}$, and $\cot=-\sqrt3$. Only sine and its reciprocal cosecant come out positive, because in quadrant II only the up value is positive.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to trigonometry practice questions.

Questions about the unit circle

Why is it called the unit circle?

Because its radius is one unit. That choice does the work: the hypotenuse of the reference triangle is $1$, so the ratio $\text{opposite}/\text{hypotenuse}$ is just the opposite side. The coordinates stop being ratios and become the trig values themselves.

How do I remember which quadrants are positive?

Use ASTC, often remembered as "All Students Take Calculus", reading counterclockwise from quadrant I. All six are positive in I, only Sine (and cosecant) in II, only Tangent (and cotangent) in III, only Cosine (and secant) in IV.

Why is tangent undefined at 90 degrees?

Because $\tan\theta=y/x$ and the point at $90^\circ$ is $(0,1)$, so the division is $1/0$. The same happens at $270^\circ$ where the point is $(0,-1)$. On a graph these show up as the vertical asymptotes of the tangent curve.

How do I convert between degrees and radians?

Multiply degrees by $\pi/180$ to get radians, and multiply radians by $180/\pi$ to get degrees. So $135^\circ\times\pi/180=3\pi/4$, and $7\pi/6\times180/\pi=210^\circ$. The anchor worth memorizing is that $180^\circ=\pi$ radians.