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Negative Angle Trigonometric Identities

How to use
  1. The six tabs switch between the negative-angle identities for sin⁡\sin, cos⁡\cos, tan⁡\tan, csc⁡\csc, sec⁡\sec and cot⁡\cot. Sine and cosine open an animated geometric proof; the other four open a derived card. Learn more about switching functions
  2. Drag the θ\theta slider between 10°10° and 80°80°. The scene, the step values and every number on the page recompute at the new angle. Learn more about the angle
  3. On a proof tab, press Play to run the proof step by step, or use Next › and ‹ Prev; Reset returns to the start and the speed menu sets the pace. Learn more about playing a proof
  4. The scene draws each step of the proof on the unit circle, mirroring the point at θ\theta across the xx-axis; the log beside it names the rule each step uses. Learn more about the scene
  5. On the tan, csc, sec and cot tabs, the derived card shows the short algebra from the sine and cosine results; the See … proof → buttons jump to the source identity. Learn more about derived identities
  6. The formula table under the tool lists all six identities with their values at the current angle; click a row to open that identity. Learn more about the formula table
  7. The value cards compare both sides of the identity at the chosen angle — they must agree, for example sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta. Learn more about the numerical check


sin(−θ) = −sin θ
θ40°
Negative angle scenexyOsin θPθP: y = sin θ−sin θP'−θP': y = −sin θy = sin(−θ)
Step 0 of 3

sin θ

0.643

sin(−θ)

-0.643

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

Key Terms
Switching Between Functions
Adjusting the Angle θ
Playing Through a Geometric Proof
Reading the Geometric Scene
Working with Derived Identities
Reading the Formula Table
Verifying Identities Numerically
Geometric Proofs: sin(-θ) and cos(-θ)
Derived Identities: tan, csc, sec, cot
Why Negative Angle Identities Matter
Related Concepts and Tools
The Sine Negative-Angle Identity
The Cosine Negative-Angle Identity
The Tangent Negative-Angle Identity
The Cosecant Negative-Angle Identity
The Secant Negative-Angle Identity
The Cotangent Negative-Angle Identity
Sine Proof, Step 1: Place P at Angle θ
Sine Proof, Step 2: Mirror P Across the x-Axis
Sine Proof, Step 3: Read Off sin(-θ)
Cosine Proof, Step 1: Place P at Angle θ
Cosine Proof, Step 2: Mirror P Across the x-Axis
Cosine Proof, Step 3: Read Off cos(-θ)






Key Terms

• Negative angle identity — a formula relating a trig function evaluated at −θ-\theta to the same function at θ\theta.
• Even function — satisfies f(−x)=f(x)f(-x) = f(x). Its graph is symmetric about the y-axis. cos⁡\cos and sec⁡\sec are even.
• Odd function — satisfies f(−x)=−f(x)f(-x) = -f(x). Its graph is symmetric about the origin. sin⁡\sin, tan⁡\tan, csc⁡\csc, cot⁡\cot are odd.
• Reflection across the x-axis — the geometric operation taking the terminal point P=(cos⁡θ,sin⁡θ)P = (\cos\theta, \sin\theta) to P′=(cos⁡θ,−sin⁡θ)P' = (\cos\theta, -\sin\theta), which is the terminal point of −θ-\theta.
• Parity — whether a function is even or odd, summarized in the Parity column of the formula table.

Switching Between Functions

Six tabs at the top let you select which negative-angle identity to study: sin⁡(−θ)\sin(-\theta), cos⁡(−θ)\cos(-\theta), tan⁡(−θ)\tan(-\theta), csc⁡(−θ)\csc(-\theta), sec⁡(−θ)\sec(-\theta), cot⁡(−θ)\cot(-\theta).

How selection changes the view:
• sin⁡\sin and cos⁡\cos open the geometric proof scene showing the reflection P→P′P \to P' on the unit circle.
• tan⁡\tan, csc⁡\csc, sec⁡\sec, and cot⁡\cot open the derived identity card with the algebraic chain.
• The active tab is highlighted in deep blue.
• The URL updates with ?negFn=...?negFn=... so links you share preserve the selected function.

Clicking any row of the formula table at the bottom also jumps to that function.

Adjusting the Angle θ

Each view exposes a slider for the angle θ\theta in degrees, between 15°15° and 75°75°.

What changes as you slide:
• On geometric scenes, PP moves along the upper unit circle and P′P' follows below as its mirror image.
• The coordinate readouts at PP and P′P' update in real time.
• The verification cards at the bottom recompute both sin⁡θ\sin\theta and sin⁡(−θ)\sin(-\theta) (or cos⁡θ\cos\theta and cos⁡(−θ)\cos(-\theta)) at the new θ\theta.

Sweep the slider to see that sin⁡(−θ)\sin(-\theta) and −sin⁡θ-\sin\theta track together (odd behavior), while cos⁡(−θ)\cos(-\theta) and cos⁡θ\cos\theta stay identical (even behavior).

Playing Through a Geometric Proof

DemoThe sine proof, step by step
Step 0 of 5
When sin⁡\sin or cos⁡\cos is active, an animated proof unfolds in three steps. A toolbar gives you control:

• Reset — return to step 0 with a blank scene.
• Prev / Next — step one stage at a time.
• Play / Pause — advance automatically.
• Speed selector — 0.5×0.5\times, 1×1\times, 1.5×1.5\times, 2×2\times.

The three steps are: (1) place P at angle θ, (2) reflect across the x-axis to produce P′P' at angle −θ-\theta, (3) read off the identity from the coordinates of P′P'.

Reading the Geometric Scene

The SVG shows the unit circle with two terminal points:

• P at angle θ\theta above the x-axis, with coordinates (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta).
• P' at angle −θ-\theta below the x-axis, with coordinates (cos⁡θ,−sin⁡θ)(\cos\theta, -\sin\theta).

Reflection across the x-axis is the key operation:
• Preserves the x-coordinate — so cosine comes out even, cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta.
• Flips the sign of the y-coordinate — so sine comes out odd, sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta.

A comparison overlay highlights the shared horizontal foot and the equal-magnitude, opposite-sign vertical legs.

Working with Derived Identities

DemoDerived identities
Step 0 of 5
Selecting tan⁡(−θ)\tan(-\theta), csc⁡(−θ)\csc(-\theta), sec⁡(−θ)\sec(-\theta), or cot⁡(−θ)\cot(-\theta) opens a different card layout. Instead of a unit-circle picture, it shows the algebraic derivation as a chain of equations.

Layout of the derived card:
• A short intro explains which earlier identity the current one rests on.
• Jump buttons link directly to the geometric proofs of sin⁡(−θ)\sin(-\theta) or cos⁡(−θ)\cos(-\theta).
• A multi-line derivation block shows each manipulation with a brief side note.
• Verification cards confirm both sides match numerically.

Two derived identities preserve sign (sec⁡\sec, like cos⁡\cos, is even); four flip sign (tan⁡\tan, csc⁡\csc, cot⁡\cot, like sin⁡\sin, are odd).

Reading the Formula Table

DemoThe formula table
Step 0 of 5
A reference table beneath every scene lists all six negative-angle identities at once:

• Function column — the trig function with −θ-\theta argument.
• Identity column — the right-hand side.
• Parity column — labels each as even or odd.
• Value column — the numeric value at the current θ\theta.
• Source column — labels each as geometric (sin⁡\sin, cos⁡\cos) or via X for the derived forms.

Click any row to make that function active. The current row gets a deep-blue left border and tinted background.

Verifying Identities Numerically

Every scene includes two metric cards that compute the function at +θ+\theta and at −θ-\theta for the active row.

Example for sin⁡\sin:
• Left card shows sin⁡θ\sin\theta.
• Right card shows sin⁡(−θ)\sin(-\theta).

For odd functions, the two values are equal in magnitude and opposite in sign. For even functions, they are identical. Sweeping the slider while watching the cards is a fast empirical check across all θ\theta, and the formula table mirrors this across all six functions at once.

Geometric Proofs: sin(-θ) and cos(-θ)

The two foundational identities come from reflecting the terminal point across the x-axis.

sin(-θ) = -sin θ — reflection flips the y-coordinate:
P=(cos⁡θ,sin⁡θ)  →  P′=(cos⁡θ,−sin⁡θ)P = (\cos\theta, \sin\theta) \;\to\; P' = (\cos\theta, -\sin\theta)

Since the y-coordinate of P′P' is sin⁡(−θ)\sin(-\theta) by definition, sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta. Sine is odd.

cos(-θ) = cos θ — reflection preserves the x-coordinate. The x-coordinate of P′P' equals the x-coordinate of PP, which is cos⁡θ\cos\theta. Therefore cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta. Cosine is even.

Each is treated stage by stage below: the sine form and the cosine form.

For full coverage with proofs in all quadrants, see the negative angle identities theory page.

Derived Identities: tan, csc, sec, cot

The four remaining identities follow from the two geometric ones:

tan(-θ) = -tan θ (odd) — tan⁡=sin⁡/cos⁡\tan = \sin/\cos:
tan⁡(−θ)=sin⁡(−θ)cos⁡(−θ)=−sin⁡θcos⁡θ=−tan⁡θ\tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin\theta}{\cos\theta} = -\tan\theta


csc(-θ) = -csc θ (odd) — reciprocal of an odd function is odd: csc⁡(−θ)=1/sin⁡(−θ)=−1/sin⁡θ=−csc⁡θ\csc(-\theta) = 1/\sin(-\theta) = -1/\sin\theta = -\csc\theta.

sec(-θ) = sec θ (even) — reciprocal of an even function is even: sec⁡(−θ)=1/cos⁡(−θ)=1/cos⁡θ=sec⁡θ\sec(-\theta) = 1/\cos(-\theta) = 1/\cos\theta = \sec\theta.

cot(-θ) = -cot θ (odd) — reciprocal of tangent: cot⁡(−θ)=1/tan⁡(−θ)=−1/tan⁡θ=−cot⁡θ\cot(-\theta) = 1/\tan(-\theta) = -1/\tan\theta = -\cot\theta.

For full derivations, see the trigonometric identities page and the reciprocal identities page.

Why Negative Angle Identities Matter

These identities reveal the symmetry structure of trigonometric functions:

• Graph symmetry — the parity rules predict whether each graph is symmetric about the y-axis (even) or about the origin (odd) without plotting points.
• Simplification — replace any f(−θ)f(-\theta) with ±f(θ)\pm f(\theta) instantly, halving the cases to consider.
• Fourier series — even functions expand into cosines only, odd functions into sines only.
• Integration — odd integrands over symmetric intervals like [−a,a][-a, a] integrate to zero.
• Solving equations — paired solutions θ\theta and −θ-\theta are predictable from parity.

All five uses come down to one of two facts, each proved in a single picture: sine flips sign, cosine does not.

For applications and examples, see the trigonometric identities applications page.

Related Concepts and Tools

Continue exploring with these connected resources:

• Pythagorean Identities — companion identities relating sin⁡2\sin^2 and cos⁡2\cos^2.
• Double Angle Identities — formulas for sin⁡(2θ)\sin(2\theta), cos⁡(2θ)\cos(2\theta) that combine with parity rules.
• Half Angle Identities — formulas for sin⁡(α/2)\sin(\alpha/2) and friends.
• Sum and Difference Identities — additive companions; combine with parity for full flexibility.
• Unit Circle — geometric setup for the reflection used in this tool.
• Trigonometric Functions Graphs — see the parity visually in each function's graph.
• Supplementary Angle Identities — the other reflection, across the yy-axis, and how it combines with parity.
• Basic Trigonometric Identities — the reciprocal and quotient identities that carry parity from sine and cosine to the other four functions.

The Sine Negative-Angle Identity

The identity sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta is proved by reflecting the terminal point across the x-axis and reading its new y-coordinate.
xyOsin θPθP: y = sin θ−sin θP′−θP′: y = −sin θy = sin(−θ)sin(−θ) = −sin θ · −0.643 = −0.643
The complete sine proof, frozen

P and its mirror image P′, with the two amber legs of equal length pointing opposite ways — the whole content of sin(−θ) = −sin θ.

Its three stages: place the point, mirror it, and read off the y-coordinate.

Sine is the odd function of the pair, and three of the four derived identities inherit their sign flip from it: cosecant directly, tangent through the quotient, and cotangent through tangent.

The Cosine Negative-Angle Identity

The identity cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta comes from the same reflection, read along the other axis: mirroring across the x-axis cannot change an x-coordinate.
xyOcos θPθP: x = cos θP′−θP′: x = cos θx = cos(−θ)cos(−θ) = cos θ · 0.766 = 0.766
The complete cosine proof, frozen

One indigo leg serving both points: mirroring across the x-axis cannot move a foot that sits on the x-axis.

Its three stages: place the point, mirror it, and read off the x-coordinate.

Under the cosine tab the tool hides the vertical legs entirely and draws a faint connector between PP and P′P' instead — the picture is arguing that the two points sit on one vertical line, which is exactly what "same x-coordinate" means. Cosine is the even function of the pair, and secant is the only derived identity that inherits that evenness.

The Tangent Negative-Angle Identity

Tangent is a quotient of one odd function and one even function, so exactly one sign flips and the quotient comes out odd.
tan(−θ) = −tan θtan(−θ)=sin(−θ) / cos(−θ)definition=(−sin θ) / (cos θ)sin(−θ) = −sin θ, cos(−θ) = cos θ=−tan θsimplifyat θ = 40° : tan(−θ) = −0.839 = −tan θ
tan(−θ), derived

Odd over even: the numerator flips, the denominator does not, so the quotient flips.

tan⁡(−θ)=sin⁡(−θ)cos⁡(−θ)=−sin⁡θcos⁡θ=−tan⁡θ\tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin\theta}{\cos\theta} = -\tan\theta


This is the general rule in miniature: odd divided by even is odd. Both ingredients are proved geometrically — the sine identity supplies the minus sign, the cosine identity supplies the unchanged denominator — and the card's jump buttons lead to each. Cotangent then inherits the flip from tangent.

The Cosecant Negative-Angle Identity

Cosecant is the reciprocal of sine, and a reciprocal keeps the parity of what it inverts.
csc(−θ) = −csc θcsc(−θ)=1 / sin(−θ)definition=1 / (−sin θ)sin(−θ) = −sin θ=−csc θsimplifyat θ = 40° : csc(−θ) = −1.556 = −csc θ
csc(−θ), derived

The minus sign travels straight out of the denominator — a reciprocal keeps its parity.

csc⁡(−θ)=1sin⁡(−θ)=1−sin⁡θ=−csc⁡θ\csc(-\theta) = \frac{1}{\sin(-\theta)} = \frac{1}{-\sin\theta} = -\csc\theta


The minus sign moves out of the denominator untouched, so cosecant is odd for the same reason sine is. Note what the identity does not fix: cosecant is undefined wherever sin⁡θ=0\sin\theta = 0, and negating the angle does not rescue it — if one side is undefined, so is the other. The slider's 15°15°–75°75° range stays clear of those angles.

The Secant Negative-Angle Identity

Secant is the reciprocal of cosine, so it is the one derived identity with no sign change at all.
sec(−θ) = sec θsec(−θ)=1 / cos(−θ)definition=1 / cos θcos(−θ) = cos θ=sec θsimplifyat θ = 40° : sec(−θ) = 1.305 = sec θ
sec(−θ), derived

The one derived card with no sign anywhere in it, because cosine had none to give.

sec⁡(−θ)=1cos⁡(−θ)=1cos⁡θ=sec⁡θ\sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos\theta} = \sec\theta


Nothing flips because nothing flipped in the cosine identity it rests on. Secant and cosine are the two even functions in the table; the other four are odd. Sweeping the slider is the quickest way to see it — the two verification cards for secant never separate, while the cards for every odd function stay opposite in sign.

The Cotangent Negative-Angle Identity

Cotangent is the reciprocal of tangent, so it flips sign for the same reason tangent does — one step further removed from the geometry.
cot(−θ) = −cot θcot(−θ)=1 / tan(−θ)definition=1 / (−tan θ)tan(−θ) = −tan θ=−cot θsimplifyat θ = 40° : cot(−θ) = −1.192 = −cot θ
cot(−θ), derived

Two steps from the picture: cotangent inverts tangent, which already inherited the flip from sine.

cot⁡(−θ)=1tan⁡(−θ)=1−tan⁡θ=−cot⁡θ\cot(-\theta) = \frac{1}{\tan(-\theta)} = \frac{1}{-\tan\theta} = -\cot\theta


Cotangent sits two derivations from the picture: it depends on tangent, which depends on sine and cosine. It can also be read straight off the definition cot⁡θ=cos⁡θ/sin⁡θ\cot\theta = \cos\theta / \sin\theta — even over odd, which is odd — and the two routes agree, as they must.

Sine Proof, Step 1: Place P at Angle θ

The sine proof opens with a single point: PP on the unit circle at angle θ\theta above the x-axis, with the vertical leg from the axis up to PP drawn in amber.
xyOsin θPθP: y = sin θ
Step 1: place P at angle θ

One point, one amber leg. On a unit circle that leg’s signed length is the y-coordinate itself.

That leg has signed length sin⁡θ\sin\theta, and because the circle has radius 11 the length is the y-coordinate of PP — no scaling in between. The red arc at the origin marks the angle θ\theta measured counter-clockwise, the positive direction.

Under this tab the tool runs in its sine-only mode, hiding the horizontal cos⁡θ\cos\theta leg so that nothing competes with the quantity being tracked. The cosine proof hides the opposite one.

Sine Proof, Step 2: Mirror P Across the x-Axis

Reflecting PP across the x-axis produces P′P', sitting directly below at angle −θ-\theta — the same rotation measured clockwise.
xyOsin θPθP: y = sin θ−sin θP′−θP′: y = −sin θ
Step 2: mirror across the x-axis

P′ appears directly below, same distance from the axis, opposite side — equal magnitude, flipped sign.

The reflection is what carries the whole argument, and it does two things at once: it leaves the x-coordinate exactly where it was, and it flips the sign of the y-coordinate. The second amber leg has the same length as the first but points the other way, so its signed length is −sin⁡θ-\sin\theta.

Nothing has been proved yet — so far this is only a construction. The claim arrives when the picture is read as a statement about the angle −θ-\theta, which is the next step.

Sine Proof, Step 3: Read Off sin(-θ)

P′P' is the terminal point of the angle −θ-\theta on the unit circle, so by the definition of sine its y-coordinate is sin⁡(−θ)\sin(-\theta).
xyOsin θPθP: y = sin θ−sin θP′−θP′: y = −sin θy = sin(−θ)sin(−θ) = −sin θ · −0.643 = −0.643
Step 3: read off sin(−θ)

The label y = sin(−θ) names the coordinate the picture already gave as −sin θ.

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


The proof is a matter of naming the same number twice. The picture gives that y-coordinate as −sin⁡θ-\sin\theta; the definition gives it as sin⁡(−θ)\sin(-\theta); therefore the two are equal. At the tool's opening angle of 40°40° both verification cards read −0.643-0.643.

That single sign flip is the definition of an odd function, and it propagates through tangent, cosecant and cotangent.

Cosine Proof, Step 1: Place P at Angle θ

The cosine proof begins from the same point PP at angle θ\theta, but with the indigo horizontal leg drawn instead of the vertical one.
xyOcos θPθP: x = cos θ
Step 1: place P at angle θ

Same point, indigo leg instead: the horizontal distance from the y-axis, equal to cos θ.

That leg runs from the origin to the foot of PP on the x-axis, and its length is cos⁡θ\cos\theta — again equal to the coordinate itself, because the radius is 11.

The tool switches to its cosine-only mode here, hiding the vertical legs and the right-angle markers that the sine proof relies on. Same circle, same point, different quantity in view.

Cosine Proof, Step 2: Mirror P Across the x-Axis

The same reflection places P′P' at angle −θ-\theta, and a faint vertical connector is drawn between PP and P′P'.
xyOcos θPθP: x = cos θP′−θP′: x = cos θ
Step 2: mirror across the x-axis

The faint connector is the argument — a vertical segment means one shared x-coordinate.

That connector is the argument. Two points joined by a vertical segment have the same x-coordinate by definition, and the horizontal indigo leg — drawn once — serves both of them. The reflection moved the point without moving its foot.

Compare with the sine version of this step: the same construction supports both proofs, and the tool draws whichever consequence the active tab is about.

Cosine Proof, Step 3: Read Off cos(-θ)

P′P' is the terminal point of −θ-\theta, so its x-coordinate is cos⁡(−θ)\cos(-\theta) — and that x-coordinate is the one PP already had.
xyOcos θPθP: x = cos θP′−θP′: x = cos θx = cos(−θ)cos(−θ) = cos θ · 0.766 = 0.766
Step 3: read off cos(−θ)

x = cos(−θ) labels the same foot that was already labelled cos θ. No sign appears.

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


No sign appears anywhere, which is the whole content of the result: cosine is even. At 40°40° both verification cards read 0.7660.766, and they stay locked together for every angle the slider reaches.

Only secant inherits this evenness; the remaining three derived identities take their sign flip from sine instead.