A row of six tabs at the top lets you select which double-angle identity to study: sin(2θ), cos(2θ), tan(2θ), csc(2θ), sec(2θ), cot(2θ).
How selection changes the view: • sin and cos open the geometric proof scene with a step-by-step animation. • 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 ?fn=..., so links you copy preserve the selected function.
You can also click any row of the formula table at the bottom to jump directly to that function.
Adjusting the Angle θ
Each view exposes a slider for the base angle θ in degrees, between 10° and 80°.
What changes as you slide: • On geometric scenes, the SVG triangle resizes and reshapes in real time. • The number readout next to the slider shows the exact degree value. • The verification cards at the bottom recompute both sides of the identity using the new θ.
Slow sweeps near 45° are useful for seeing how the relationships behave in the most symmetric case, while values near the extremes (10° or 80°) show how the same identities still hold for narrow and wide triangles.
Playing Through a Geometric Proof
When sin or cos is active, an animated proof unfolds in six steps. A toolbar gives you control:
• Reset — return to step 0 with a blank scene. • Prev / Next — step through one stage at a time. • Play / Pause — advance automatically. • Speed selector — 0.5×, 1×, 1.5×, or 2×.
Each step adds one geometric element (radii, triangle fill, bisector, half-angles, leg labels, final metrics). The right-hand panel logs each step's name and rationale, so you can stop and re-read at any point.
Reading the Geometric Scene
The SVG shows the unit circle with two radii OA and OB of length 1 meeting at the center O with angle 2θ between them.
Elements that appear across the steps: • Red arc at O — the apex angle, labeled C=2θ. • Indigo chordAB — the base of the isosceles triangle. • Blue segmentOM — the perpendicular bisector, equal to cosθ. • Half-angles at O — each labeled θ once the bisector is drawn. • Half-chord labels — each labeled sinθ on segments MA and MB.
A small right-angle mark appears at M when the perpendicular bisector becomes visible.
Working with Derived Identities
Selecting tan(2θ), csc(2θ), sec(2θ), or cot(2θ) opens a different card layout. Instead of a triangle, the page 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 the source identities. • A multi-line derivation block shows each manipulation with a brief side note. • Verification cards confirm both sides match numerically.
This split keeps the geometric ideas isolated to two functions and treats the other four as quick algebraic consequences.
Reading the Formula Table
A reference table beneath every scene lists all six identities at once:
• Function column — the name of the trig function with 2θ argument. • Identity column — the right-hand side of the formula. • Value column — the numeric value computed at the current θ. • Source column — labels each identity as geometric (sin, cos) or via X for derived ones.
Click any row to make that function active. The current selection gets a deep-blue left border and a tinted background, making it easy to track context.
Verifying Identities Numerically
Every scene includes two metric cards near the bottom that compute both sides of the active identity at the current θ.
Example for sin(2θ): • Left card shows sin(2θ). • Right card shows 2sinθcosθ.
The two numbers always match (within rounding to three decimals). Sweeping the slider while watching the cards is a fast empirical check that the identity holds for every angle, not just the one in the picture. The formula table mirrors this behavior across all six functions simultaneously.
Geometric Proofs: sin(2θ) and cos(2θ)
The two foundational identities are proved by drawing an isosceles triangle with two unit radii.
sin(2θ) = 2 sin θ cos θ — area is computed two ways:
area=21sin(2θ)=sinθcosθ
Multiplying by 2 gives the identity.
cos(2θ) = 1 - 2 sin²θ — the law of cosines gives ∣AB∣2=2−2cos(2θ), while the half-chord computation gives ∣AB∣2=4sin2θ. Equating the two yields the result.
For full coverage of these proofs and equivalent forms, see the double angle identities theory page.
The four remaining identities follow directly from the first two:
tan(2θ) — from tan=sin/cos:
tan(2θ)=1−tan2θ2tanθ
csc(2θ) — reciprocal of sin(2θ):
csc(2θ)=2sinθcosθ1
sec(2θ) — reciprocal of cos(2θ):
sec(2θ)=1−2sin2θ1
cot(2θ) — reciprocal of tan(2θ):
cot(2θ)=2tanθ1−tan2θ
For step-by-step derivations of each, see the trigonometric identities page and the reciprocal identities page.
Why Double-Angle Identities Matter
Double-angle identities show up across mathematics and physics:
• Integration — sin2θ and cos2θ become integrable after substituting cos(2θ)=1−2sin2θ or 2cos2θ−1. • Equation solving — equations mixing sinθ with sin(2θ) collapse to single-angle equations after substitution. • Wave physics and signal processing — sums of sinusoids reduce via these formulas, separating frequency components. • Geometry and circular motion — relating arc, chord, and apothem in regular polygons uses sin(2θ) and cos(2θ) directly.
For applications and worked examples, see the trigonometric identities applications page.
Related Concepts and Tools
Continue exploring with these connected resources:
• Pythagorean Identities — sin2θ+cos2θ=1 and its companions. • Sum and Difference Identities — sin(α±β) and cos(α±β), from which double-angle identities follow as the case α=β. • Half-Angle Identities — solve the double-angle formulas backward to express sin(θ/2) and cos(θ/2). • Unit Circle — geometric setup for every identity in this tool. • Trigonometric Functions Graphs — see how sin, cos, tan and their reciprocals evolve as θ varies. • Triangle Explorer — interactive triangles with built-in law of sines and law of cosines.