Unit Circle Trig Definitions

Unit Circle Trig Definitions

A set of definitions of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent

Unit circle on x-y axes with point (x,y) and angle θ. Definitions: sinθ=y, cosθ=x, tanθ=y/x, cscθ=1/y, secθ=1/x, cotθ=x/y.

See also

Circle trig definitions, SOHCAHTOA

Key Formula

\cos\theta = x, \quad \sin\theta = y, \quad \tan\theta = \frac{y}{x}

\sec\theta = \frac{1}{x}, \quad \csc\theta = \frac{1}{y}, \quad \cot\theta = \frac{x}{y}

Where:

$\theta$ = The angle measured from the positive x-axis (counterclockwise is positive)
$x$ = The x-coordinate of the point where the terminal side of θ intersects the unit circle
$y$ = The y-coordinate of the point where the terminal side of θ intersects the unit circle

Worked Example

Problem: Use the unit circle to find all six trig function values at θ = π/3 (60°).

Step 1: Identify the point on the unit circle at θ = π/3. From the standard unit circle, this point is:

\left(\frac{1}{2},\; \frac{\sqrt{3}}{2}\right)

Step 2: Read off cosine and sine directly from the coordinates:

\cos\frac{\pi}{3} = \frac{1}{2}, \quad \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}

Step 3: Compute tangent as y/x:

\tan\frac{\pi}{3} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}

Step 4: Compute the three reciprocal functions:

\sec\frac{\pi}{3} = \frac{1}{1/2} = 2, \quad \csc\frac{\pi}{3} = \frac{1}{\sqrt{3}/2} = \frac{2\sqrt{3}}{3}, \quad \cot\frac{\pi}{3} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Answer: cos(π/3) = 1/2, sin(π/3) = √3/2, tan(π/3) = √3, sec(π/3) = 2, csc(π/3) = 2√3/3, cot(π/3) = √3/3.

Another Example

This example uses a quadrantal angle (on an axis) where the x-coordinate is 0, showing that some trig functions become undefined. The first example used a standard angle inside a quadrant where all six functions have finite values.

Problem: Find all six trig function values at θ = 3π/2 (270°) using the unit circle.

Step 1: Locate the point on the unit circle at θ = 3π/2. This angle points straight down, so the point is:

(0,\; -1)

Step 2: Read off cosine and sine from the coordinates:

\cos\frac{3\pi}{2} = 0, \quad \sin\frac{3\pi}{2} = -1

Step 3: Compute tangent. Since x = 0, the denominator is zero, so tangent is undefined:

\tan\frac{3\pi}{2} = \frac{-1}{0} = \text{undefined}

Step 4: Compute the reciprocal functions. Secant requires dividing by x = 0, so it is also undefined. Cosecant and cotangent use y = −1:

\sec\frac{3\pi}{2} = \text{undefined}, \quad \csc\frac{3\pi}{2} = \frac{1}{-1} = -1, \quad \cot\frac{3\pi}{2} = \frac{0}{-1} = 0

Answer: cos(3π/2) = 0, sin(3π/2) = −1, tan(3π/2) = undefined, sec(3π/2) = undefined, csc(3π/2) = −1, cot(3π/2) = 0.

Frequently Asked Questions

What is the difference between unit circle trig definitions and SOHCAHTOA?

SOHCAHTOA defines sine, cosine, and tangent using the sides of a right triangle and only works for acute angles (between 0° and 90°). The unit circle definitions use coordinates on a circle of radius 1, which naturally extends the trig functions to all angles — including obtuse angles, negative angles, and angles greater than 360°. For acute angles, both approaches give the same results.

Why is the unit circle important for trigonometry?

The unit circle provides a single, unified framework for defining all six trig functions at every possible angle. It makes it easy to determine the sign of each function in each quadrant, to see periodicity (the values repeat every 2π), and to identify angles where functions are undefined. Nearly all advanced topics — such as trig identities, graphing trig functions, and calculus with trig — rely on the unit circle definitions.

How do you remember the coordinates on the unit circle?

For the key first-quadrant angles, note that the cosine values at 0°, 30°, 45°, 60°, and 90° follow the pattern √4/2, √3/2, √2/2, √1/2, √0/2 (which simplify to 1, √3/2, √2/2, 1/2, 0). The sine values follow the same pattern in reverse order. For other quadrants, use symmetry and adjust signs based on which quadrant the angle falls in.

Unit Circle Trig Definitions vs. SOHCAHTOA (Right Triangle Definitions)

	Unit Circle Trig Definitions	SOHCAHTOA (Right Triangle Definitions)
Basis	Coordinates (x, y) on a circle of radius 1	Ratios of sides (opposite, adjacent, hypotenuse) of a right triangle
Valid angles	All real angles (any θ)	Only acute angles (0° < θ < 90°)
Sine definition	sin θ = y-coordinate	sin θ = opposite / hypotenuse
Cosine definition	cos θ = x-coordinate	cos θ = adjacent / hypotenuse
Handles negative values	Yes — coordinates can be negative	No — side lengths are always positive
When to use	General trig, graphing, calculus, any angle	Quick calculations with acute angles in triangles

Why It Matters

The unit circle trig definitions appear throughout precalculus, calculus, and physics. When you graph sine or cosine waves, evaluate limits of trig functions, or decompose vectors into components, you are using these definitions. Mastering the unit circle also gives you instant recall of exact trig values at standard angles — a skill tested heavily on the SAT, ACT, and AP exams.

Common Mistakes

Mistake: Mixing up the coordinate order: thinking sin θ is the x-coordinate and cos θ is the y-coordinate.

Correction: Remember that cosine corresponds to the x-coordinate and sine corresponds to the y-coordinate. An alphabetical trick: 'c' (cosine) comes before 's' (sine), just as 'x' comes before 'y'.

Mistake: Forgetting to check whether x or y equals zero before computing tangent, secant, cosecant, or cotangent.

Correction: At quadrantal angles (0°, 90°, 180°, 270°), one coordinate is zero. Dividing by zero makes the function undefined, not zero. Always verify the denominator before computing a ratio.

Related Terms

Trig Functions — General overview of all six trig functions
Sine — The y-coordinate on the unit circle
Cosine — The x-coordinate on the unit circle
Tangent — Ratio sin θ / cos θ, or y/x on the unit circle
Cosecant — Reciprocal of sine, equals 1/y
Secant — Reciprocal of cosine, equals 1/x
Cotangent — Reciprocal of tangent, equals x/y
SOHCAHTOA — Right-triangle trig definitions for acute angles
Circle Trig Definitions — Trig definitions using a circle of any radius