The Unit Circle: Radians, Reference Angles and Exact Trig Values

What the unit circle is

The unit circle is the circle of radius 1 centred at the origin, with equation $x^2 + y^2 = 1$. Angles are measured from the positive x-axis, counterclockwise for positive angles and clockwise for negative ones. The side where an angle starts is called the initial side and the side where it ends is the terminal side.

For any angle $\theta$, the terminal side meets the circle at exactly one point. The definition that makes the unit circle so useful is that this point has coordinates $(\cos\theta, \sin\theta)$. Tangent is the ratio of the two:

For acute angles this agrees with the right-triangle definitions you may already know. Drop a perpendicular from the point to the x-axis and you get a right triangle with hypotenuse 1, adjacent side $x$ and opposite side $y$. So $\cos\theta = \frac{x}{1}$ and $\sin\theta = \frac{y}{1}$. The unit circle extends those definitions to every angle, including obtuse, negative and very large ones.

Degrees and radians

A full turn is $360^\circ$, or $2\pi$ radians. One radian is the angle that cuts off an arc equal in length to the radius. On the unit circle, where the radius is 1, the radian measure of an angle is exactly the length of the arc it cuts off, which is why radians are the natural unit in calculus.

• $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$, $90^\circ = \frac{\pi}{2}$.
• $150^\circ = 150 \times \frac{\pi}{180} = \frac{5\pi}{6}$.
• $\frac{7\pi}{4}$ radians $= \frac{7}{4} \times 180^\circ = 315^\circ$.
• One radian is about $57.3^\circ$.

The special angles in the first quadrant

Three angles have exact values that come from two familiar triangles. The $45^\circ$ angle comes from an isosceles right triangle with hypotenuse 1, whose legs are both $\frac{\sqrt{2}}{2}$. The $30^\circ$ and $60^\circ$ angles come from half of an equilateral triangle with side 1, whose legs are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.

• $0$: point $(1, 0)$, so $\cos 0 = 1$, $\sin 0 = 0$.
• $\frac{\pi}{6}$: point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, so $\tan\frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.
• $\frac{\pi}{4}$: point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so $\tan\frac{\pi}{4} = 1$.
• $\frac{\pi}{3}$: point $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, so $\tan\frac{\pi}{3} = \sqrt{3}$.
• $\frac{\pi}{2}$: point $(0, 1)$, so $\sin\frac{\pi}{2} = 1$ and $\tan\frac{\pi}{2}$ is undefined.

A memory pattern helps: the sines of $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$ are $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$, and the cosines are the same list in reverse. Every point also satisfies $x^2 + y^2 = 1$; for example, $\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1$.

Reference angles and the other quadrants

You do not need to memorise all sixteen standard points separately. Every angle has a reference angle: the acute angle between its terminal side and the x-axis. The coordinates of the angle have the same size as those of its reference angle, and only the signs change according to the quadrant.

• Quadrant I ($0$ to $\frac{\pi}{2}$): $x$ and $y$ positive, so all three functions are positive.
• Quadrant II ($\frac{\pi}{2}$ to $\pi$): $x$ negative, $y$ positive, so only sine is positive.
• Quadrant III ($\pi$ to $\frac{3\pi}{2}$): both negative, so only tangent is positive.
• Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): $x$ positive, $y$ negative, so only cosine is positive.

Similarly, $\frac{3\pi}{4}$ has reference angle $\frac{\pi}{4}$ in quadrant II, giving the point $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ and $\tan\frac{3\pi}{4} = -1$.

Negative angles and angles beyond one turn

Adding or subtracting a full turn, $2\pi$, lands on the same point, so sine and cosine repeat every $2\pi$. Angles that share a terminal side are called coterminal. To evaluate a large angle, subtract multiples of $2\pi$ until it lies between 0 and $2\pi$; for a negative angle, add multiples of $2\pi$.

• $\frac{17\pi}{4} = 4\pi + \frac{\pi}{4}$, so it has the same point as $\frac{\pi}{4}$: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.
• $-\frac{\pi}{6}$ is coterminal with $\frac{11\pi}{6}$, so $\cos\left(-\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ and $\sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}$.
• $750^\circ = 720^\circ + 30^\circ$, so $\sin 750^\circ = \frac{1}{2}$.

Reflecting across the x-axis also shows two symmetry rules: $\cos(-\theta) = \cos\theta$, so cosine is even, and $\sin(-\theta) = -\sin\theta$, so sine is odd. Tangent has a shorter period of $\pi$, because the points for $\theta$ and $\theta + \pi$ are opposite each other and give the same ratio $\frac{y}{x}$.

The Pythagorean identity and finding other ratios

Since every point on the circle satisfies $x^2 + y^2 = 1$, every angle satisfies the most important identity in trigonometry:

The quadrant is what decides the sign; the identity alone only gives the size. The guide to trig identities builds many more identities from this one.

Solving basic trig equations with the circle

To solve an equation like $\sin\theta = \frac{1}{2}$ for $0 \le \theta < 2\pi$, look for every point on the circle with y-coordinate $\frac{1}{2}$. There are two: one in quadrant I at $\frac{\pi}{6}$ and one in quadrant II at $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$.

Likewise, $\cos\theta = -\frac{\sqrt{2}}{2}$ needs points with x-coordinate $-\frac{\sqrt{2}}{2}$, which occur at $\frac{3\pi}{4}$ in quadrant II and $\frac{5\pi}{4}$ in quadrant III. If the question asks for all solutions, add $2\pi k$ for any integer $k$: $\theta = \frac{3\pi}{4} + 2\pi k$ or $\theta = \frac{5\pi}{4} + 2\pi k$.

A calculator's inverse sine or cosine returns only one answer. The unit circle shows you where the second solution is, which calculators never do.

How to learn it efficiently

Memorising sixteen points as separate facts is slow and fragile. A better approach is to learn three things well: the three first-quadrant points for $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$; how to find a reference angle; and which signs apply in each quadrant. With those, any standard angle takes a few seconds.

Practise by drawing the circle from memory once a day for a week, labelling angles in both degrees and radians. Then test yourself with random angles, including negative ones and ones larger than $2\pi$. The unit circle calculator gives exact values and the reference angle for any input, so you can check your answers immediately. Later topics, such as the law of sines and calculus of trigonometric functions, assume this fluency.

Common mistakes

• Swapping the coordinates: the x-coordinate is cosine and the y-coordinate is sine.
• Using degree mode on a calculator for a radian question, or the reverse.
• Measuring a reference angle from the y-axis instead of the x-axis.
• Giving only one solution to an equation like $\sin\theta = \frac{1}{2}$.
• Forgetting that tangent is undefined where cosine is zero, at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$.

From the circle to the sine and cosine graphs

Imagine a point travelling counterclockwise around the unit circle at a steady speed. Its height above the x-axis rises from 0 to 1 during the first quarter turn, falls back to 0 at half a turn, drops to $-1$ at three quarters and returns to 0 after a full turn. Plotting that height against the angle produces the familiar wave of $y = \sin x$. Plotting the horizontal position instead produces $y = \cos x$, the same wave shifted a quarter turn to the left.

Everything about those graphs can be read from the circle. Their range is $[-1, 1]$ because the coordinates of a point on a circle of radius 1 never exceed 1 in size. Their period is $2\pi$ because one full turn brings the point back to where it started. The zeros of sine are at multiples of $\pi$, where the point crosses the x-axis, and the zeros of cosine are at odd multiples of $\frac{\pi}{2}$, where it crosses the y-axis. Seeing the graphs as a record of circular motion is also why trigonometric functions model tides, sound waves, alternating current and anything else that repeats in cycles.

Frequently asked questions