• Negative angle identity — a formula relating a trig function evaluated at −θ to the same function at θ. • Even function — satisfies f(−x)=f(x). Its graph is symmetric about the y-axis. cos and sec are even. • Odd function — satisfies f(−x)=−f(x). Its graph is symmetric about the origin. sin, tan, csc, cot are odd. • Reflection across the x-axis — the geometric operation taking the terminal point P=(cosθ,sinθ) to P′=(cosθ,−sinθ), which is the terminal point of −θ. • 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), cos(−θ), tan(−θ), csc(−θ), sec(−θ), cot(−θ).
How selection changes the view: • sin and cos open the geometric proof scene showing the reflection P→P′ on the unit circle. • tan, csc, sec, and cot open the derived identity card with the algebraic chain. • The active tab is highlighted in deep blue. • The URL updates with ?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 in degrees, between 15° and 75°.
What changes as you slide: • On geometric scenes, P moves along the upper unit circle and P′ follows below as its mirror image. • The coordinate readouts at P and P′ update in real time. • The verification cards at the bottom recompute both sinθ and sin(−θ) (or cosθ and cos(−θ)) at the new θ.
Sweep the slider to see that sin(−θ) and −sinθ track together (odd behavior), while cos(−θ) and cosθ stay identical (even behavior).
Playing Through a Geometric Proof
When sin or 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×, 1×, 1.5×, 2×.
The SVG shows the unit circle with two terminal points:
• P at angle θ above the x-axis, with coordinates (cosθ,sinθ). • P' at angle −θ below the x-axis, with coordinates (cosθ,−sinθ).
Reflection across the x-axis is the key operation: • Preserves the x-coordinate — so cosine comes out even, cos(−θ)=cosθ. • Flips the sign of the y-coordinate — so sine comes out odd, sin(−θ)=−sinθ.
A comparison overlay highlights the shared horizontal foot and the equal-magnitude, opposite-sign vertical legs.
Working with Derived Identities
Selecting tan(−θ), csc(−θ), sec(−θ), or cot(−θ) 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) or cos(−θ). • 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, like cos, is even); four flip sign (tan, csc, cot, like sin, are odd).
Reading the Formula Table
A reference table beneath every scene lists all six negative-angle identities at once:
• Function column — the trig function with −θ argument. • Identity column — the right-hand side. • Parity column — labels each as even or odd. • Value column — the numeric value at the current θ. • Source column — labels each as geometric (sin, 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 and at −θ for the active row.
Example for sin: • Left card shows sinθ. • Right card shows sin(−θ).
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, 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=(cosθ,sinθ)→P′=(cosθ,−sinθ)
Since the y-coordinate of P′ is sin(−θ) by definition, sin(−θ)=−sinθ. Sine is odd.
cos(-θ) = cos θ — reflection preserves the x-coordinate. The x-coordinate of P′ equals the x-coordinate of P, which is cosθ. Therefore cos(−θ)=cosθ. Cosine is even.
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) with ±f(θ) 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] integrate to zero. • Solving equations — paired solutions θ and −θ are predictable from parity.
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 sin2 and cos2. • Double Angle Identities — formulas for sin(2θ), cos(2θ) that combine with parity rules. • Half Angle Identities — formulas for sin(α/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 y-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)=−sinθ is proved by reflecting the terminal point across the x-axis and reading its new y-coordinate.
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.
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)=cosθ comes from the same reflection, read along the other axis: mirroring across the x-axis cannot change an x-coordinate.
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.
Under the cosine tab the tool hides the vertical legs entirely and draws a faint connector between P and 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), derived
Odd over even: the numerator flips, the denominator does not, so the quotient flips.
tan(−θ)=cos(−θ)sin(−θ)=cosθ−sinθ=−tanθ
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), derived
The minus sign travels straight out of the denominator — a reciprocal keeps its parity.
csc(−θ)=sin(−θ)1=−sinθ1=−cscθ
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 sinθ=0, and negating the angle does not rescue it — if one side is undefined, so is the other. The slider's 15°–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), derived
The one derived card with no sign anywhere in it, because cosine had none to give.
sec(−θ)=cos(−θ)1=cosθ1=secθ
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), derived
Two steps from the picture: cotangent inverts tangent, which already inherited the flip from sine.
cot(−θ)=tan(−θ)1=−tanθ1=−cotθ
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 cotθ=cosθ/sinθ — 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: P on the unit circle at angle θ above the x-axis, with the vertical leg from the axis up to P drawn in amber.
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 sinθ, and because the circle has radius 1 the length is the y-coordinate of P — no scaling in between. The red arc at the origin marks the angle θ measured counter-clockwise, the positive direction.
Under this tab the tool runs in its sine-only mode, hiding the horizontal cosθ 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 P across the x-axis produces P′, sitting directly below at angle −θ — the same rotation measured clockwise.
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 −sinθ.
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, which is the next step.
Sine Proof, Step 3: Read Off sin(-θ)
P′ is the terminal point of the angle −θ on the unit circle, so by the definition of sine its y-coordinate issin(−θ).
Step 3: read off sin(−θ)
The label y = sin(−θ) names the coordinate the picture already gave as −sin θ.
sin(−θ)=−sinθ
The proof is a matter of naming the same number twice. The picture gives that y-coordinate as −sinθ; the definition gives it as sin(−θ); therefore the two are equal. At the tool's opening angle of 40° both verification cards read −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 P at angle θ, but with the indigo horizontal leg drawn instead of the vertical one.
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 P on the x-axis, and its length is cosθ — again equal to the coordinate itself, because the radius is 1.
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′ at angle −θ, and a faint vertical connector is drawn between P and P′.
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′ is the terminal point of −θ, so its x-coordinate iscos(−θ) — and that x-coordinate is the one P already had.
Step 3: read off cos(−θ)
x = cos(−θ) labels the same foot that was already labelled cos θ. No sign appears.
cos(−θ)=cosθ
No sign appears anywhere, which is the whole content of the result: cosine is even. At 40° both verification cards read 0.766, 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.