Six tabs at the top let you pick which half-angle identity to study: sin(α/2), cos(α/2), tan(α/2), csc(α/2), sec(α/2), cot(α/2).
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 ?halfFn=..., so links you copy preserve the selected function.
Clicking any row of the formula table at the bottom also jumps to that function.
Adjusting the Angle α
Each view exposes a slider for the base angle $\alpha$ in degrees, between 20° and 160°. The half angle is then α/2, ranging from 10° to 80°.
What changes as you slide: • On geometric scenes, the SVG triangle resizes and the apex α updates immediately. • The number readout shows the current value of α. • The verification cards recompute both sides of the identity at the new α/2.
Sweep through several values to confirm each identity is not a coincidence at one angle but a true equality.
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×, 2×.
Each step adds one geometric element (radii, triangle fill, bisector, half-angles, leg labels, final metrics). The right-hand panel logs each step with its name and reasoning.
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 α between them.
Elements that appear across the steps: • Red arc at O — the apex angle α. • Indigo chordAB — the base of the isosceles triangle. • Blue segmentOM — the perpendicular bisector, equal to cos(α/2). • Half-angles at O — each labeled α/2 once the bisector is drawn. • Half-chord labels — sin(α/2) 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(α/2), csc(α/2), sec(α/2), or cot(α/2) opens a different card layout. Instead of a triangle, 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 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 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 α/2 argument. • Identity column — the right-hand side of the formula. • Value column — the numeric value at the current α/2. • 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 row gets a deep-blue left border and tinted background.
Verifying Identities Numerically
Every scene includes two metric cards that compute both sides of the active identity at the current α.
Example for sin(α/2): • Left card shows sin(α/2). • Right card shows (1−cosα)/2.
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 α, not just the one in the picture. The formula table mirrors this across all six functions simultaneously.
Geometric Proofs: sin(α/2) and cos(α/2)
The two foundational identities are proved by drawing an isosceles triangle with two unit radii.
sin(α/2) — apply the law of cosines and equate with the squared chord:
∣AB∣2=2−2cosα=(2sin(α/2))2
Solving gives sin(α/2)=(1−cosα)/2.
cos(α/2) — from Pythagoras in the half-triangle, cos2(α/2)=1−sin2(α/2). Substituting the sin half-angle identity:
cos(α/2)=(1+cosα)/2
For full coverage and equivalent forms (including sign choices by quadrant), see the half angle identities theory page.
The four remaining identities follow directly from the first two:
tan(α/2) — from tan=sin/cos applied to the half angle:
tan(α/2)=1+cosα1−cosα
csc(α/2) — reciprocal of sin(α/2):
csc(α/2)=1−cosα2
sec(α/2) — reciprocal of cos(α/2):
sec(α/2)=1+cosα2
cot(α/2) — reciprocal of tan(α/2):
cot(α/2)=1−cosα1+cosα
For step-by-step derivations and alternative forms, see the trigonometric identities page and the reciprocal identities page.
Why Half-Angle Identities Matter
Half-angle identities are essential whenever a problem asks for a trig function at an angle that is not on the unit circle but is half of one that is.
• Exact-value computation — find sin15°, cos22.5°, tan75° from sin30°, cos45°, cos150°. • Integration — the Weierstrass substitution t=tan(α/2) converts rational trig integrals into rational functions of t. • Equation solving — reduce equations mixing sinα and sin(α/2) to single-argument form. • Geometry — apothems, chord lengths, and inscribed-polygon side lengths all use half-angle formulas.
For applications and worked examples, see the trigonometric identities applications page.
Related Concepts and Tools
Continue exploring with these connected resources:
• Double Angle Identities — companion formulas for sin(2θ), cos(2θ), and beyond. • Pythagorean Identities — sin2θ+cos2θ=1 and its companions, used in the cos half-angle proof. • Sum and Difference Identities — base relations from which double- and half-angle identities are derived. • Unit Circle — geometric setup for every identity in this tool. • Trigonometric Functions Graphs — see how sin, cos, tan and their reciprocals evolve as the angle varies. • Triangle Explorer — interactive triangles with built-in law of sines and law of cosines.