How are sine and cosine defined on the unit circle?

On the unit circle, the terminal point of angle theta has coordinates (cos theta, sin theta). The x-coordinate is cos theta and the y-coordinate is sin theta. This definition works for any real angle, positive or negative, including angles beyond 360 degrees, and extends trigonometry beyond the right-triangle setting.

What is the difference between reciprocal and quotient identities?

Reciprocal identities express one function as 1 divided by another, like sec theta equals 1 over cos theta. Quotient identities express one function as a ratio of two others, like tan theta equals sin theta over cos theta. Reciprocal identities pair functions in three couples, while quotient identities show how the four remaining functions all build from sin and cos.

Why does tangent have a period of 180 degrees but sine has 360?

Tangent is the ratio sin theta over cos theta. When theta increases by 180 degrees, both sin and cos flip sign, and a negative divided by a negative is positive. The two sign flips cancel inside the ratio, so tan returns to its original value after only half a turn. Sine and cosine themselves need a full 360-degree turn to return.

How do you find a trig function value for any angle?

First, reduce the angle to its reference angle, which is the acute angle between the terminal ray and the x-axis. Compute or recall the function value for that acute angle. Finally, determine the sign from which quadrant the original angle terminates in. The standard ASTC pattern fixes the sign for each function in each quadrant.

Basic Trigonometric Identities

How to use. Drag the blue dot on the circle (or use the slider) to change θ. The slider sweeps through multiple turns — past 360° the arc spirals outward and the graph shows ghost dots at coterminal angles. Toggle deg/rad for units, use the tabs below to pick a function, and the Prev / Play / Next controls to step through.

sin(θ)

θ30°

Unit circle

Function graph

Step 1 of 5

sin θ

0.500

Derivation

1Place the angle

Rotate the ray from the positive x-axis through θ counterclockwise. The terminal point P sits on the unit circle.

Function

Reading

Value at θ

Getting Started

Switching Between the Six Functions

Stepping Through the Derivation

Reading the Unit Circle Display

Reading the Graph

Exploring Periodicity

The Live Formula Table

What Are the Basic Trigonometric Identities?

Defining Trig Functions on the Unit Circle

Periodicity and Reference Angles

Related Concepts and Tools

Getting Started

The explorer opens centered on the sine function. Three controls drive everything you see:

• Drag the blue dot on the circle to rotate θ
• Use the slider below the circle for precise multi-turn rotation
• Toggle deg / rad for angle units

The graph on the right plots the active function against θ in real time. As you move the dot, the terminal point P traces the unit circle and the dot on the graph tracks the function value.

Switching Between the Six Functions

Two controls swap the active function:

• The tab strip above the circle shows sin, cos, tan, csc, sec, cot in order
• The formula table below the circle is also clickable — tap any row to make it active

When you switch, the diagram redraws to show the correct leg or legs and the graph swaps to that function's curve. The URL updates with a ?fn= query parameter, so any function view can be shared as a direct link.

Stepping Through the Derivation

Each function has a five-stage derivation accessed through the Prev / Play / Next controls:

1. Place the angle — the ray rotates and P appears on the unit circle
2. Identify the leg or legs — vertical for sine, horizontal for cosine, both for tangent
3. Read the value — leg length for sin and cos, a ratio for tan and cot, a reciprocal for csc and sec
4. Sign and range — the quadrant logic and the reference angle
5. Periodicity — drag past 360° to see the spiral arc and coterminal ghost dots

Use Play to auto-advance or step manually with Prev and Next. The rule and description update at each stage.

Reading the Unit Circle Display

The unit circle on the left shows several elements that build up across the steps:

• The red ray from the origin marks the current angle

\theta

• The blue point P sits at

(\cos\theta, \sin\theta)

• The signed leg — vertical, horizontal, or both — shows the function's geometric meaning
• A small arc near the origin marks the reference angle once revealed

Watch how each leg flips sign as P crosses an axis. That sign flip is the geometric origin of all four quadrant sign rules.

Reading the Graph

The graph on the right plots the active function across multiple periods. Key features to watch:

• A blue dot tracks the function value at the current θ
• Vertical dashed lines mark asymptotes — at 90° and 270° for tan and sec, at 0°, 180°, and 360° for cot and csc
• Ghost dots highlight coterminal angles where the function takes the same value

The curve's range is fixed per function: bounded between

-1

and

1

for

\sin\theta

and

\cos\theta

, unbounded for the other four.

Exploring Periodicity

The slider sweeps past 360° to reveal what makes trig functions cyclic. As θ exceeds one full turn:

• The ray keeps rotating but P returns to the same circle position
• The arc near the origin spirals outward to count the rotations
• Ghost dots appear on the graph at every coterminal angle

For sin, cos, csc, and sec, this means

f(\theta + 360°) = f(\theta)

. For tan and cot, the period is shorter —

f(\theta + 180°) = f(\theta)

— visible as twice the repetition rate on the graph.

The Live Formula Table

Below the circle, a table lists all six functions with their current values at θ:

• The Function column names the function with

(\theta)

notation
• The Reading column states the geometric or algebraic recipe — vertical leg,

\sin\theta / \cos\theta

1 / \sin\theta

, and so on
• The Value at θ column updates as you drag

Watch how reciprocal pairs move together: when

\sin\theta

is small,

\csc\theta

is large. When

\cos\theta = 0

\sec\theta

blows up. The table makes these relationships numerical and concrete.

What Are the Basic Trigonometric Identities?

The basic identities relate the six trig functions to each other through simple ratios. They fall into two groups.

Reciprocal identities:

\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

Quotient identities:

\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Together they mean only two of the six functions — usually

\sin\theta

and

\cos\theta

— are truly independent. The other four are algebraic combinations of those two.

For broader coverage, see trigonometric identities theory.

Defining Trig Functions on the Unit Circle

On the unit circle, the terminal point of angle θ sits at coordinates

(\cos\theta, \sin\theta)

. From this single fact, every trig function follows:

•

\sin\theta

is the y-coordinate of P
•

\cos\theta

is the x-coordinate of P
•

\tan\theta = \sin\theta / \cos\theta

is the slope of the ray
•

\csc\theta

\sec\theta

\cot\theta

are reciprocals of the above

This definition extends trig beyond right triangles to any real angle — positive or negative, more than 360°, in any quadrant. For the right-triangle perspective, see right triangle trigonometry.

Periodicity and Reference Angles

Two facts make trig functions usable for any angle, no matter how large.

Periodicity means values repeat at a fixed interval. Sin, cos, csc, and sec have period

360°

(or

2\pi

). Tan and cot have period

180°

(or

\pi

) because dividing two functions that both flip sign produces a function that does not.

Reference angle is the acute angle between the terminal ray and the x-axis. It reduces any angle to a Q1 calculation — once you know

|\sin 30°| = 0.5

, you know

|\sin\theta| = 0.5

for every coterminal or reflected angle. The quadrant supplies the sign.

For deeper coverage, see periodicity and reference angles.

Related Concepts and Tools

Explore connected topics:

Trigonometric functions graphs — adjustable amplitude, period, and phase shift for each function

Functions signs by quadrant — focused view of the four quadrant sign patterns

Angle types explorer — acute, obtuse, reflex, and coterminal classification

Pythagorean identity — the

\sin^2\theta + \cos^2\theta = 1

relationship from the unit circle

Double angle identities — geometric derivations for

\sin 2\theta

and

\cos 2\theta