Six tabs at the top let you pick which Pythagorean form to study: sinθ, cosθ, tanθ, cscθ, secθ, cotθ.
How selection changes the view: • sin and cos open the geometric proof scene with a step-by-step animation built on a right triangle inside 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 ?fn=... so links you share preserve the selected function.
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 10° and 80° (first quadrant).
What changes as you slide: • On geometric scenes, the triangle inside the unit circle reshapes in real time. • The number readout shows the exact degree value. • The verification cards at the bottom recompute both sides of the identity at the new θ.
Restricting to the first quadrant keeps every trig function positive, which lets the tool take square roots without sign ambiguity.
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 element (radii, triangle fill, bisector, half-angles, leg labels sinθ and cosθ, final metrics). The right panel logs each step with reasoning.
Reading the Geometric Scene
The SVG shows the unit circle with two radii OA and OB meeting at the center O, with a perpendicular bisector OM.
Elements that appear across the steps: • Indigo chordAB — the base of the isosceles triangle. • Blue segmentOM — the bisector, equal to cosθ in right triangle OMA. • Half-chord labels — sinθ on segment MA. • Right-angle mark at M — the key to applying Pythagoras.
Once the legs are labeled, the identity sin2θ+cos2θ=1 follows from leg² + leg² = hypotenuse² with hypotenuse 1.
Working with Derived Identities
Selecting tanθ, cscθ, secθ, or cotθ 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 manipulation (divide by sin2θ or cos2θ) produces the identity. • Jump buttons link directly to the geometric proofs of the source identities sinθ and cosθ. • A multi-line derivation block shows each manipulation with a side note. • Verification cards confirm both sides match numerically.
This split keeps the geometric idea isolated to the base identity and treats the other four as algebraic consequences.
Reading the Formula Table
A reference table beneath every scene lists all six Pythagorean identities at once:
• Function column — the active trig function. • Identity column — the identity expressed as a square root. • Value column — the numeric value at the current θ. • Source column — labels each as geometric (sinθ, cosθ) or via sin, cos 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 θ.
Example for sinθ: • Left card shows sinθ. • Right card shows 1−cos2θ.
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 θ in the first quadrant. The formula table mirrors this across all six functions simultaneously.
Geometric Proof: sin²θ + cos²θ = 1
The base identity is proved directly from a right triangle inscribed in the unit circle.
In right triangle OMA: • Hypotenuse OA=1 (a radius of the unit circle). • Leg OM=cosθ. • Leg MA=sinθ.
Applying Pythagoras:
sin2θ+cos2θ=1
Solving for sinθ or cosθ and taking the positive root gives the two geometric identities:
sinθ=1−cos2θ,cosθ=1−sin2θ
For full coverage and equivalent forms across all quadrants, see the Pythagorean identities theory page.
Derived Identities: tan, sec, csc, cot
The four remaining identities follow by dividing the base by sin2θ or cos2θ.
Dividing sin2θ+cos2θ=1 by cos2θ:
tan2θ+1=sec2θ
This gives tanθ=sec2θ−1 and secθ=1+tan2θ.
Dividing sin2θ+cos2θ=1 by sin2θ:
1+cot2θ=csc2θ
This gives cotθ=csc2θ−1 and cscθ=1+cot2θ.
For step-by-step derivations and the unsigned forms valid in all quadrants, see the trigonometric identities page and the reciprocal identities page.
Why Pythagorean Identities Matter
The three Pythagorean identities are the most-used identities in trigonometry:
• Simplification — convert expressions in sin2 to cos2 form and vice versa. • Integration — u-substitution in integrals like ∫sec2θdθ=tanθ+C relies on 1+tan2θ=sec2θ. • Equation solving — quadratic equations in sinθ or cosθ frequently emerge after substitution. • Proofs of other identities — sum, difference, double-angle, and half-angle identities all use Pythagorean relations along the way.
For applications and worked examples, see the trigonometric identities applications page.
Related Concepts and Tools
Continue exploring with these connected resources:
• Double Angle Identities — formulas for sin(2θ), cos(2θ), tan(2θ) built on Pythagoras. • Half Angle Identities — formulas for sin(α/2) and friends, derived using sin2+cos2=1. • Sum and Difference Identities — additive companions to Pythagoras. • 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 Pythagoras verification.