The Unit Circle

Idea

The triangle got you this far, but it has an edge it cannot cross. At 0, 90, 180, 270 and 360 degrees the angle lands on an axis. There is no triangle there, so the right-triangle definitions cannot help.

The circle can. Draw a circle of radius $1$ and let the angle $x$ sweep from the positive horizontal axis. The angle lands on the point $(\cos x, \sin x)$ - that point is the definition from now on, and it works at every angle, axis or not. Calculus reads sine and cosine off this circle, and so will the rest of this module.

Rule

Cosine is the first coordinate, sine the second. Read them straight off the point.

$(\cos 0^\circ, \sin 0^\circ) = (1, 0)$

$(\cos 90^\circ, \sin 90^\circ) = (0, 1)$

$(\cos 180^\circ, \sin 180^\circ) = (-1, 0)$

$(\cos 270^\circ, \sin 270^\circ) = (0, -1)$

$(\cos 360^\circ, \sin 360^\circ) = (1, 0)$

One full turn ends where it began, so $0^\circ$ and $360^\circ$ share a point. That is periodicity: $\sin(x + 360^\circ) = \sin x$ for every angle.

How it is used

Angle	Radians	Point	$\cos$	$\sin$	$\tan$	$\cot$
$0^\circ$	$0$	$(1, 0)$	$1$	$0$	$0$	undefined
$90^\circ$	$\dfrac{\pi}{2}$	$(0, 1)$	$0$	$1$	undefined	$0$
$180^\circ$	$\pi$	$(-1, 0)$	$-1$	$0$	$0$	undefined
$270^\circ$	$\dfrac{3\pi}{2}$	$(0, -1)$	$0$	$-1$	undefined	$0$
$360^\circ$	$2\pi$	$(1, 0)$	$1$	$0$	$0$	undefined

Why tangent and cotangent break in opposite places

Neither is a coordinate. Both are built from the two coordinates, and each fails when its own bottom is $0$ .

\tan x = \frac{\sin x}{\cos x}

\cot x = \frac{\cos x}{\sin x}

At	$\tan$	$\cot$
$0^\circ$ , $180^\circ$ , $360^\circ$	$0$	undefined
$90^\circ$ , $270^\circ$	undefined	$0$

Where one is zero the other is undefined. On a graph these are the vertical asymptotes of each curve.

The special angles land here too

The values from the last lesson have not gone anywhere. At $30^\circ$ , $45^\circ$ and $60^\circ$ the point on this circle is $(\cos x, \sin x)$ with the exact values you already know - the hypotenuse is $1$ , so each leg is its own ratio. Check them in the Explore below.

Worked example

Evaluate $\tan\pi + \cos 2\pi + \sin\dfrac{3\pi}{2}$ .

Remember

$\pi = 180^\circ$ , half a turn. $2\pi = 360^\circ$ , a full one degrees $\rightarrow$ radians: $\times \dfrac{\pi}{180}$ radians $\rightarrow$ degrees: $\times \dfrac{180}{\pi}$ $\dfrac{3\pi}{2} \rightarrow \dfrac{3\pi}{2} \cdot \dfrac{180}{\pi} = 270^\circ$

Convert, since the circle is labelled in degrees.

\tan 180^\circ + \cos 360^\circ + \sin 270^\circ

Tangent is not a coordinate. Build it from the point at $180^\circ$ , which is $(-1, 0)$ .

\underbrace{\tan 180^\circ}_{\frac{\sin 180^\circ}{\cos 180^\circ} = \frac{0}{-1}} = 0

The other two are coordinates. At $360^\circ$ the point is $(1, 0)$ , at $270^\circ$ it is $(0, -1)$ .

0 + \underbrace{\cos 360^\circ}_{1} + \underbrace{\sin 270^\circ}_{-1}

Add the three.

0 + 1 - 1 = 0

A zero on the bottom breaks a quotient. A zero on the top does not, which is why $\tan 180^\circ$ is $0$ rather than undefined.

The unit circle

Try this. Try 0°, 90°, 180° and 270°. The horizontal projection is cosine; the vertical projection is sine. Track their signs between axes.

Angle (degrees)270

θ = 270° = 1.5π rad · cos θ = 0 · sin θ = -1 · sin² θ + cos² θ = 1

The unit circle

Try this. Try 0°, 90°, 180° and 270°. The horizontal projection is cosine; the vertical projection is sine. Track their signs between axes.

Angle (degrees)180

θ = 180° = 1π rad · cos θ = -1 · sin θ = 0

Explore

Drag from $60$ degrees up to $90$ . The cos leg shrinks to nothing, and tan - sine divided by that shrinking number - grows huge, then reads undefined at $90$ exactly. Cross to $105$ and tan returns, large and negative.

Then drag to $360$ : the readings are $0$ 's again. One full turn changes nothing. On the way, stop at $30$ , $45$ and $60$ and check the readouts against the table you learned last lesson.

The unit circle

Try this. Try 0°, 90°, 180° and 270°. The horizontal projection is cosine; the vertical projection is sine. Track their signs between axes.

Angle (degrees)60

θ = 60° = 0.333π rad · cos θ = 0.5 · sin θ = 0.866

Idea

Rule

How it is used

Why tangent and cotangent break in opposite places

The special angles land here too

Worked example

Try it yourself.

Explore

Pause before the next idea.