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Trigonometric Functions
and the Unit Circle

Discover how the unit circle extends \(\sin\), \(\cos\), and \(\tan\) beyond acute angles to all real numbers, using radians as measures of rotation.

Precalculus

What Is the Unit Circle?

The unit circle is a circle centered at the origin with radius \(r = 1\). Every point on it satisfies the equation \(x^2 + y^2 = 1\). It lives in the standard coordinate plane and gives us a geometric home for trigonometric functions.

x y 1 -1 1 -1 O (x, y) r = 1
\(x^2 + y^2 = 1\)

Radians: Angles as Arc Length

A radian measures an angle by the length of the arc it subtends on the unit circle. One full revolution is \(2\pi\) radians. Because the circumference of a unit circle is \(2\pi \cdot 1 = 2\pi\), radians and arc length are the same number.

\(\pi/6\)
\(= 30^\circ\)
\(\pi/4\)
\(= 45^\circ\)
\(\pi/3\)
\(= 60^\circ\)
\(\pi/2\)
\(= 90^\circ\)

Why the Unit Circle Matters

Right-triangle trigonometry only works for acute angles (between \(0^\circ\) and \(90^\circ\)). But angles in physics, engineering, and calculus can be any real number: negative, greater than \(360^\circ\), or even irrational multiples of \(\pi\). The unit circle removes all restrictions.

1
Right-triangle definition: limited to \(0 < \theta < \frac{\pi}{2}\).
2
Unit circle definition: works for \(\theta \in (-\infty, \infty)\).
3
Result: \(\sin\) and \(\cos\) become functions of all real numbers.

The Core Definition

Let \(\theta\) be a real number. Starting from the point \((1, 0)\), travel counterclockwise around the unit circle an arc of length \(|\theta|\) (clockwise if \(\theta < 0\)). You arrive at a point \(P = (x, y)\). Then:

(1, 0) θ P(x, y)
\(\cos\theta = x\)
\(\sin\theta = y\)
 CCW

Signs Across Four Quadrants

As the terminal point moves around the circle, the signs of \(x\) and \(y\) change. This determines whether \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) are positive or negative.

Q I
\(\sin +, \cos +\)
\(\tan +\)
 Q II
\(\sin +, \cos -\)
\(\tan -\)
 Q III
\(\sin -, \cos -\)
\(\tan +\)
 Q IV
\(\sin -, \cos +\)
\(\tan -\)
 0
\(\frac{\pi}{2}\)
\(\pi\)
\(\frac{3\pi}{2}\)

Degrees vs Radians

Both measure angles, but they use different units. In the unit circle framework, radians are the natural choice because they directly equal arc length.

Property Degrees Radians
Full revolution \(360^\circ\) \(2\pi\)
Right angle \(90^\circ\) \(\pi/2\)
Straight angle \(180^\circ\) \(\pi\)
Arc length on unit circle Not direct Equal to angle measure
Used in calculus Rarely Always

Reference Angles and Symmetry

A reference angle is the acute angle between the terminal side and the \(x\)-axis. It lets us reduce any angle to a first-quadrant calculation, then apply the correct sign.

1
Q I: reference angle = \(\theta\)
2
Q II: reference angle = \(\pi - \theta\)
3
Q III: reference angle = \(\theta - \pi\)
4
Q IV: reference angle = \(2\pi - \theta\)

Step-by-Step Method

To evaluate \(\sin\theta\) or \(\cos\theta\) for any real number \(\theta\):

1
Coterminal angle: reduce \(\theta\) to \([0, 2\pi)\) by adding or subtracting \(2\pi\).
2
Identify the quadrant where the terminal side lands.
3
Find the reference angle (acute angle to the \(x\)-axis).
4
Evaluate \(\sin\) and \(\cos\) of the reference angle (use known values).
5
Apply the sign based on the quadrant (ASTC mnemonic).

Example 1: Evaluate \(\sin\!\left(\dfrac{5\pi}{6}\right)\)

P
\(\frac{5\pi}{6}\)
\(\frac{\pi}{6}\)
\(\sin\theta = \frac{1}{2}\)
 (1,0)
1
\(\frac{5\pi}{6}\) is in Quadrant II (between \(\frac{\pi}{2}\) and \(\pi\)).
2
Reference angle: \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
3
\(\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}\). Sine is positive in Q II.
4
Result: \(\sin\!\left(\frac{5\pi}{6}\right) = \frac{1}{2}\).

Example 2: Evaluate \(\cos\!\left(\dfrac{7\pi}{4}\right)\)

P
\(\frac{7\pi}{4}\)
\(\frac{\pi}{4}\)
\(\cos\theta = \frac{\sqrt{2}}{2}\)
1
\(\frac{7\pi}{4}\) is in Quadrant IV (between \(\frac{3\pi}{2}\) and \(2\pi\)).
2
Reference angle: \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\).
3
\(\cos\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). Cosine is positive in Q IV.
4
Result: \(\cos\!\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Example 3: A Negative Angle

Evaluate \(\tan\!\left(-\dfrac{\pi}{3}\right)\).

1
Negative angle means rotate clockwise by \(\frac{\pi}{3}\).
2
Coterminal angle: \(-\frac{\pi}{3} + 2\pi = \frac{5\pi}{3}\), which is in Q IV.
3
Reference angle: \(2\pi - \frac{5\pi}{3} = \frac{\pi}{3}\).
4
\(\tan\!\left(\frac{\pi}{3}\right) = \sqrt{3}\). Tangent is negative in Q IV.
5
Result: \(\tan\!\left(-\frac{\pi}{3}\right) = -\sqrt{3}\).

Periodicity of Sine and Cosine

Because the terminal point returns to the same position after each full revolution, the trig functions repeat:

\(\sin\) and \(\cos\)
Period = \(2\pi\)
Domain: all real numbers
Range: \([-1, 1]\)
\(\tan\)
Period = \(\pi\)
Domain: \(\theta \neq \frac{\pi}{2} + k\pi\)
Range: \((-\infty, \infty)\)

Graphing \(\sin\theta\) and \(\cos\theta\)

\(\sin\theta\) (green)
Starts at 0, peaks at \(\pi/2\), crosses 0 at \(\pi\).
\(\cos\theta\) (indigo)
Starts at 1, crosses 0 at \(\pi/2\), troughs at \(\pi\).

Common Mistakes

Wrong sign
Forgetting to check the quadrant. e.g., writing \(\cos\!\left(\frac{2\pi}{3}\right) = \frac{1}{2}\) instead of \(-\frac{1}{2}\).
Degrees in formulas
Plugging 90 instead of \(\frac{\pi}{2}\) into radian-based equations. Calculus requires radians.
Reference angle errors
Using \(\theta - \frac{\pi}{2}\) instead of \(\pi - \theta\) in Q II, or \(\theta\) instead of \(\theta - \pi\) in Q III.

Key Values Summary

\(\theta\) \(\sin\theta\) \(\cos\theta\) \(\tan\theta\)
\(0\) \(0\) \(1\) \(0\)
\(\pi/6\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\)
\(\pi/4\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) \(1\)
\(\pi/3\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\)
\(\pi/2\) \(1\) \(0\) undefined
\(\pi\) \(0\) \(-1\) \(0\)
\(3\pi/2\) \(-1\) \(0\) undefined

Question 1 of 5

What is the value of \(\sin\!\left(\dfrac{3\pi}{4}\right)\)?

Question 2 of 5

On the unit circle, the point corresponding to \(\theta = \pi\) has coordinates:

Question 3 of 5

What is \(\cos\!\left(\dfrac{5\pi}{3}\right)\)?

Question 4 of 5

The period of the tangent function is:

Question 5 of 5

If \(\theta = -\dfrac{\pi}{2}\), what are the coordinates of the terminal point on the unit circle?