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Supplementary Angle Trigonometric Identities


sin(π − θ) = sin θ
θ35°
Reflection sceneLegendcos θsin θcos (reflected)sin (reflected)y-axis mirrorθ anglegap to mirror (×2)x-axisy-axis90°PP'Oθ = 35°gap to mirror90° − 35°same gap (mirrored)90° − 35°new angle180° − 35°
Step 0 of 6

sin(π − θ)

0.574

sin θ

0.574

Derivation
Press Play to step through the proof.
Function
Identity
Sign
Value
Source

Getting Started With the Explorer
Switching Between Geometric and Derived Views
Reading the Identity Bar and Derivation Steps
Tracing Identities Back to Their Source
Using the Formula Comparison Table
Verifying Identities Numerically
What Are Supplementary Angle Identities?
Why Reflection Across the y-Axis?
The Six Identities at a Glance
Supplementary vs Complementary Identities
Related Concepts and Tools






Getting Started With the Explorer

Use the tabs at the top to switch between the six trig functions evaluated at π−θ\pi - \theta. Each tab loads either a geometric reflection view (for sin and cos) or a derivation card (for tan, csc, sec, and cot). The θ\theta slider below the visualization controls the angle from 15∘15^\circ to 75∘75^\circ, and every number, formula, and visual updates in real time as you drag.

The formula comparison table at the bottom stays visible across all tabs. It shows every identity, its sign behavior, and the current numerical value side by side, so the relationships between the six functions are always in view.

Your current tab is preserved in the URL through a supFnsupFn query parameter. Refresh the page or share the link and the same function loads back up.

Switching Between Geometric and Derived Views

The six identities split into two groups based on how they are proved:

sin and cos open the geometric reflection view. A unit-circle scene shows θ\theta and its supplementary counterpart π−θ\pi - \theta as mirror images across the y-axis, with the relevant coordinate highlighted to make the sign behavior visible.

tan, csc, sec, and cot open a derivation card. The identity bar at the top states the result, a three-step derivation expands it from the definition, and metric cards verify it numerically at the current θ\theta.

The tabs themselves carry the function name in italic and the (π−θ)(\pi - \theta) argument in upright type. The active tab fills with deep blue; inactive tabs sit in the neutral tab-strip background. Click any tab to switch instantly.

Reading the Identity Bar and Derivation Steps

Each derivation card opens with the identity bar: a single coloured equation summarizing the result. The argument π−θ\pi - \theta is rendered in red. The source ratio on the right of the equals sign is coloured to match its parent: deep blue for sin-derived identities, amber for cos-derived ones.

Below the bar, the derivation panel lays out the proof in three rows:

Row 1 expands the function into its definition (for example, tan⁡(π−θ)=sin⁡(π−θ)/cos⁡(π−θ)\tan(\pi - \theta) = \sin(\pi - \theta) / \cos(\pi - \theta)).

Row 2 substitutes the two root supplementary identities: sin stays the same, cos flips sign.

Row 3 simplifies to the final form.

A small note in faint gray on the right of each row spells out which rule was applied at that step. Read top to bottom to follow the algebra; the layout matches a textbook proof.

Tracing Identities Back to Their Source

The four derived identities — tan, csc, sec, and cot — each carry one or two See [source] proof buttons under the introductory text. Clicking one jumps the active tab directly to that parent geometric identity:

tan and cot pull from both sin and cos

csc pulls from sin alone

sec pulls from cos alone

When a source button is clicked, the reflection scene reloads with the new function highlighted, and the URL updates so back-button navigation works as expected. Use these buttons to walk a full algebraic chain end-to-end — start at the result, follow the citations back to the unit-circle reflection, then click the next tab to come back. The whole tour takes about a minute and makes the dependency structure of the six identities concrete.

Using the Formula Comparison Table

The bottom formula table is a clickable directory of all six supplementary identities. Each row shows five fields:

Function — the function name with its (π−θ)(\pi - \theta) argument

Identity — the simplified right-hand side of the equation

Sign — a flips or unchanged badge (red for flips, blue for unchanged)

Value — the numerical value at the current θ\theta, in tabular numbers so columns align across rows

Source — either geometric for sin and cos, or via sin, via cos, or via sin, cos for the four derived identities

Click any row to jump to that function. The active row marks itself with a deep-blue left border and a tinted background, so the current tab is always identifiable at a glance, even while scrolling.

Verifying Identities Numerically

Drag the $\theta$ slider to confirm each identity holds across the 15∘15^\circ to 75∘75^\circ range. In every derivation card, two metric cards display the left-hand side and right-hand side of the identity computed independently at the current θ\theta. The two values should match to three decimal places at every slider position.

This is especially useful when sign behavior feels counterintuitive. For example, at θ=60∘\theta = 60^\circ:

cos⁡(π−60∘)=cos⁡(120∘)=−0.500\cos(\pi - 60^\circ) = \cos(120^\circ) = -0.500


−cos⁡(60∘)=−0.500-\cos(60^\circ) = -0.500


Both panels display −0.500-0.500, confirming the identity. Try sweeping the slider across the full range and watch the two values move together while keeping their equality. The bottom table values update in lock-step, so any discrepancy would be immediately visible.

What Are Supplementary Angle Identities?

Supplementary angles are two angles whose measures sum to π\pi radians (180∘180^\circ). For any angle θ\theta, its supplementary partner is π−θ\pi - \theta. The supplementary identities express how each of the six trig functions behaves when its input changes from θ\theta to π−θ\pi - \theta.

Two functions stay unchanged: sin and csc.

Four functions flip sign: cos, tan, sec, and cot.

The split is not arbitrary. It mirrors a reflection of the unit circle point across the vertical axis: the y-coordinate (which gives sin) is preserved; the x-coordinate (which gives cos) negates. Every other identity inherits its behavior from this single geometric fact.

For broader context, see the trigonometric identities reference.

Why Reflection Across the y-Axis?

On the unit circle, an angle θ\theta corresponds to a point P=(cos⁡θ,sin⁡θ)P = (\cos\theta, \sin\theta). The supplementary angle π−θ\pi - \theta corresponds to a second point P′P' that sits at the reflection of $P$ across the y-axis.

Reflection across the vertical axis negates the x-coordinate but leaves the y-coordinate alone. Since sin⁡θ=y\sin\theta = y and cos⁡θ=x\cos\theta = x:

sin⁡(π−θ)=sin⁡θ\sin(\pi - \theta) = \sin\theta


cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos\theta


Every other identity then follows by algebra. Tan, csc, sec, and cot are built from sin and cos by definition, so the sign behavior of those two building blocks fully determines all four. The tool's geometric view shows this reflection directly; the derivation cards show the algebra.

For more on the underlying geometry, see the unit circle visualizer.

The Six Identities at a Glance

All six supplementary angle identities in one place:

sin⁡(π−θ)=sin⁡θ\sin(\pi - \theta) = \sin\theta


cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos\theta


tan⁡(π−θ)=−tan⁡θ\tan(\pi - \theta) = -\tan\theta


csc⁡(π−θ)=csc⁡θ\csc(\pi - \theta) = \csc\theta


sec⁡(π−θ)=−sec⁡θ\sec(\pi - \theta) = -\sec\theta


cot⁡(π−θ)=−cot⁡θ\cot(\pi - \theta) = -\cot\theta


Notice the pairing: each function and its reciprocal share sign behavior. Sin and csc are both unchanged. Cos and sec both flip. Tan and cot both flip. This makes sense algebraically — taking a reciprocal cannot introduce or remove a sign — and reduces memorization to three rules instead of six.

For a comprehensive reference covering all related identity families, see the trigonometric identities page.

Supplementary vs Complementary Identities

Supplementary and complementary identities are easy to confuse but produce different results.

Supplementary angles sum to π\pi (180∘180^\circ): the partner angle is π−θ\pi - \theta. Geometrically, this is reflection across the y-axis, which preserves sin and flips cos.

Complementary angles sum to π/2\pi/2 (90∘90^\circ): the partner angle is π/2−θ\pi/2 - \theta. Geometrically, this is reflection across the line y=xy = x, which swaps sin and cos entirely.

So for sin⁡\sin: the supplementary identity gives sin⁡θ\sin\theta, but the complementary identity gives cos⁡θ\cos\theta. For cos⁡\cos: supplementary gives −cos⁡θ-\cos\theta, while complementary gives sin⁡θ\sin\theta. The two sets are not interchangeable.

For the partner family, see the complementary angle identities visualizer.

Related Concepts and Tools

Related Concepts:

Complementary Angle Identities — Partner identity family covering the case π/2−θ\pi/2 - \theta.

Opposite Angle Identities — Behavior of trig functions at −θ-\theta (reflection across the x-axis).

Double Angle Identities — Identities for 2θ2\theta that build on the supplementary and complementary results.

Half Angle Identities — Identities for θ/2\theta/2, completing the elementary identity family.

Unit Circle Visualizer — The geometric foundation behind every reflection identity.

Trigonometric Function Graphs — Plots that show the symmetries underlying these identities visually.

Trig Identities Reference — Comprehensive collection of all major identity families with proofs.