Use the tabs at the top to switch between the six trig functions evaluated at π−θ. Each tab loads either a geometric reflection view (for sin and cos) or a derivation card (for tan, csc, sec, and cot). The θ slider below the visualization controls the angle from 15∘ to 75∘, 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 supFn 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 and its supplementary counterpart π−θ as mirror images across the y-axis, with the relevant coordinate highlighted to make the sign behavior visible. Both tabs run the same six-stage reflection proof.
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.
The tabs themselves carry the function name in italic and the (π−θ) 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 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)).
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:
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) 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, 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 θ slider to confirm each identity holds across the 15∘ to 75∘ 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. 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∘:
cos(π−60∘)=cos(120∘)=−0.500
−cos(60∘)=−0.500
Both panels display −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 πradians (180∘). For any angle θ, its supplementary partner is π−θ. The supplementary identities express how each of the six trig functions behaves when its input changes from θ to π−θ.
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 corresponds to a point P=(cosθ,sinθ). The supplementary angle π−θ corresponds to a second point 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 sinθ=y and cosθ=x:
sin(π−θ)=sinθ
cos(π−θ)=−cosθ
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)=sinθ
cos(π−θ)=−cosθ
tan(π−θ)=−tanθ
csc(π−θ)=cscθ
sec(π−θ)=−secθ
cot(π−θ)=−cotθ
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 (180∘): the partner angle is π−θ. Geometrically, this is reflection across the y-axis, which preserves sin and flips cos.
Complementary angles sum to π/2 (90∘): the partner angle is π/2−θ. Geometrically, this is reflection across the line y=x, which swaps sin and cos entirely.
So for sin: the supplementary identity gives sinθ, but the complementary identity gives cosθ. For cos: supplementary gives −cosθ, while complementary gives sinθ. 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.
Because sine is the y-coordinate and the mirror is vertical, nothing about the height changes — which is why this identity carries no minus sign. The cosecant identity inherits that directly, being sine's reciprocal. At θ=35° both verification cards read 0.574.
The Cosine Supplementary Identity
The identity cos(π−θ)=−cosθ comes from the same reflection read along the horizontal: mirroring across the y-axis negates every x-coordinate.
The same scene read for cosine, frozen
The blue leg runs right, the teal one runs left by the same amount — equal length, opposite direction.
It rests on the same six stages as the sine identity — the tool runs one scene for both tabs and swaps only the identity bar and the two metric cards.
In the frozen picture the two horizontal legs are drawn in different colours for exactly this reason: the deep-blue leg runs right from the origin, the teal one runs left by the same distance. Same length, opposite direction, hence the minus sign. Secant, tangent and cotangent all take their sign flip from here. At θ=35° both cards read −0.819.
The Tangent Supplementary Identity
Tangent divides a quantity that survives the reflection by one that flips, so the quotient flips.
tan(π − θ), derived
Unchanged numerator over flipped denominator, so the quotient flips.
tan(π−θ)=cos(π−θ)sin(π−θ)=−cosθsinθ=−tanθ
Both ingredients are proved geometrically — sine unchanged, cosine negated — and the card's two jump buttons lead to each. Cotangent is the same statement upside down. At θ=35° the identity reads −0.700 on both sides.
The Cosecant Supplementary Identity
Cosecant is the reciprocal of sine, and since sine is unchanged at π−θ, cosecant is unchanged too.
csc(π − θ), derived
Sine passes through the reflection untouched, and its reciprocal does the same.
csc(π−θ)=sin(π−θ)1=sinθ1=cscθ
Together with sine this is one of only two identities on the page with no sign change; the other four all flip. Taking a reciprocal can never introduce or remove a minus sign, which is why the six identities reduce to three rules. At θ=35° both sides read 1.743.
The Secant Supplementary Identity
Secant inverts cosine, so it inherits cosine's sign flip exactly.
sec(π − θ), derived
The minus sign comes straight out of the denominator, inherited from cosine.
sec(π−θ)=cos(π−θ)1=−cosθ1=−secθ
Its single source is the cosine identity, which is why this card shows only one jump button where tangent shows two. At θ=35° both sides read −1.221.
The Cotangent Supplementary Identity
Cotangent is cosine over sine — the flipped quantity over the unchanged one — so it flips as well.
cot(π − θ), derived
Flipped numerator over unchanged denominator — the mirror image of the tangent card.
cot(π−θ)=sin(π−θ)cos(π−θ)=sinθ−cosθ=−cotθ
Read as the reciprocal of tangent the answer is the same, as it must be. Either route traces back to the same two geometric facts: sine unchanged, cosine negated. At θ=35° both sides read −1.428.
Reflection Proof, Step 1: Setup
The proof opens with a single point P on the unit circle at angle θ from the x-axis, with both legs of its right triangle drawn: the horizontal leg is cosθ, the vertical leg is sinθ, and the hypotenuseOP has length 1.
Step 1: the unit triangle at θ
Horizontal leg cos θ in blue, vertical leg sin θ in brown, hypotenuse fixed at 1.
The two legs are coloured differently throughout the tool — deep blue for the horizontal, brown for the vertical — because the whole argument turns on the two behaving differently under the mirror.
The y-axis is drawn in as the mirror line, marked with the arc showing that its angle from the x-axis is exactly 90°.
Step 2: the y-axis as mirror
The mirror gets its own angle marked: 90° from the x-axis.
Naming the mirror's own angle matters, because the next two steps measure everything relative to it. The choice of the y-axis is what makes this the supplementary family — reflecting across the line y=x instead produces the complementary identities, and reflecting across the x-axis produces the negative-angle ones.
Reflection Proof, Step 3: Measure the Gap to the Mirror
The amber arc measures the angular distance from OP up to the mirror: since OP sits at θ and the mirror sits at 90°, that gap is 90°−θ.
Step 3: the gap to the mirror
The amber arc measures 90° − θ, which at 35° is 55° — the quantity reflection must preserve.
At the frozen angle of 35° the gap is 55°. This quantity is the hinge of the argument — it is the one thing a reflection is guaranteed to preserve.
Note that the gap is measured as an angle, not a distance. That is what lets the proof work with any θ the slider reaches rather than one particular triangle.
Reflection Proof, Step 4: Reflect Across the Mirror
Reflecting P across the y-axis produces P′, and the second amber arc shows the reflected gap: the same 90°−θ, now on the far side of the mirror.
Step 4: reflect
The second amber arc matches the first, and the mirrored triangle appears in teal and violet.
That equality is the definition of a reflection — it preserves distances and angles to the mirror line — and the tool flags it with a callout at this step.
The reflected triangle appears in its own colours: teal for the horizontal leg, violet for the vertical one. Comparing them with the originals is what the final step does.
Reflection Proof, Step 5: Read the New Angle
Adding the mirror's own 90° to the reflected gap of 90°−θ gives the angle of OP′ measured from the positive x-axis.
Step 5: the new angle
The green arc totals 90° + 55° = 145°, which is 180° − θ: the supplementary angle itself.
90°+(90°−θ)=180°−θ
The green arc traces that total. At θ=35° it reads 145°. In radians this is π−θ — so P′ is not merely some mirrored point, it is the terminal point of the supplementary angle. Everything before this step was construction; this is where the construction acquires a name.
Reflection Proof, Step 6: Compare Coordinates
The last step names the same point twice and equates the two descriptions.
Step 6: compare coordinates
One point named twice — once by reflection, once as the terminal point of π − θ.
From the reflection, P′=(−cosθ,sinθ). From the previous step, P′ is the terminal point of π−θ, so P′=(cos(π−θ),sin(π−θ)). Matching coordinates:
cos(π−θ)=−cosθsin(π−θ)=sinθ
Both geometric identities fall out of this single comparison, which is why the tool runs one animation for two tabs. The four derived identities are algebra on top of these two lines.