Trigonometric Identities as Operations on Ratios

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Identities are operations on ratios — not strings to memorize

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Part 21 of the Learning Quantum Physics series.

Trigonometry is the study of ratios of a triangle’s sides. Every textbook “identity” is an operation on those ratios — square, add, subtract, flip, compose, take a square root. The formula is the measurement of that meaning, not a string to memorize.

This handbook is the sequel to Six Trigonometric Functions — One Diagram, which draws \(\cos\theta\), \(\sin\theta\), \(\tan\theta\), \(\cot\theta\), \(\sec\theta\), and \(\csc\theta\) as six colored segments on one unit circle. For radius, radian, and the rest of the circle vocabulary, see Anatomy of a Circle.

Here each family is one operation: what it means, a figure, the boxed formulas, and a one-line closer.

1. Ratios, then operations

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On the unit circle of that article, the radius is \(1\), so each segment equals its function value — not merely “proportional to” it.

1. Orange \(OW\) on the \(x\)-axis is \(\cos\theta\).
2. Green \(WP\) up to \(P\) is \(\sin\theta\).
3. Purple \(VT\) on the vertical tangent \(x=1\) is \(\tan\theta\).
4. Cyan dashed ray \(OT\) is \(\sec\theta\).
5. Pink \(RC\) on the horizontal tangent \(y=1\) is \(\cot\theta\).
6. Magenta solid ray \(OC\) is \(\csc\theta\).

Point \(P=(\cos\theta,\sin\theta)\) is where the angle ray meets the circle. Reciprocal pairs multiply to \(1\) because they sit on similar triangles, not because someone flipped a symbol. The rest of this article asks: what can you do to those lengths?

Operation	Meaning	Family
Square + add	Two legs make hypotenuse 1	§2
Reflect	Height flips sign	§3
Swap legs	Complement	§4
Add a full turn	Same point \(P\)	§5
Compose two angles	Mix the two pairs	§6
Square + subtract	Double the angle	§7
Square root	Half the angle	§8
Multiply heights	Sum of two waves	§9
Scale + add	One shifted wave	§10

The hero figure is that same circle plus a map of the operations. Jump to the family you need; each section below is one row of the table.

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2. Pythagorean — square the legs, add

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The two legs of \(\triangle OWP\) and the hypotenuse \(OP=1\) already satisfy Pythagoras. Square the orange run, square the green rise, add: the sum is the square of the radius.

\[ \boxed{\sin^2\theta+\cos^2\theta=1} \]

\[ \boxed{1+\tan^2\theta=\sec^2\theta} \]

\[ \boxed{1+\cot^2\theta=\csc^2\theta} \]

The last two are the same statement read on the tangent triangles of Six Trigonometric Functions — One Diagram: divide the first identity by \(\cos^2\theta\) and the purple/cyan pair appears; divide by \(\sin^2\theta\) and the pink/magenta pair appears. Squaring and adding did not invent three facts — it measured one right triangle three ways.

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3. Odd, even, and allied angles — reflect the ray

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Reflect the ray over the \(x\)-axis. The orange width stays; the green height flips sign. That is all “odd” and “even” mean on this diagram. Allied angles are the same idea at the other axes: fold the ray onto a first-quadrant copy and keep track of the sign.

\[ \boxed{\sin(-\theta)=-\sin\theta,\quad \csc(-\theta)=-\csc\theta} \]

\[ \boxed{\cos(-\theta)=\cos\theta,\quad \sec(-\theta)=\sec\theta} \]

\[ \boxed{\tan(-\theta)=-\tan\theta,\quad \cot(-\theta)=-\cot\theta} \]

Sine, cosecant, tangent, and cotangent are odd. Cosine and secant are even. Reflecting across the \(y\)-axis instead (the supplement) keeps the green height and flips the orange width:

\[ \boxed{\sin(\pi-\theta)=\sin\theta,\qquad \cos(\pi-\theta)=-\cos\theta} \]

The compact allied-angle table is the same reflection, written at every axis. Tangent is the ratio \(\sin/\cos\), so it does not need its own memorization grid.

Angle	\(\sin\)	\(\cos\)	\(\tan=\sin/\cos\)
\(\pi/2-\theta\)	\(\cos\theta\)	\(\sin\theta\)	\(\cot\theta\)
\(\pi/2+\theta\)	\(\cos\theta\)	\(-\sin\theta\)	\(-\cot\theta\)
\(\pi-\theta\)	\(\sin\theta\)	\(-\cos\theta\)	\(-\tan\theta\)
\(\pi+\theta\)	\(-\sin\theta\)	\(-\cos\theta\)	\(\tan\theta\)
\(3\pi/2-\theta\)	\(-\cos\theta\)	\(-\sin\theta\)	\(\cot\theta\)
\(3\pi/2+\theta\)	\(-\cos\theta\)	\(\sin\theta\)	\(-\cot\theta\)
\(2\pi-\theta\)	\(-\sin\theta\)	\(\cos\theta\)	\(-\tan\theta\)
\(2\pi+\theta\)	\(\sin\theta\)	\(\cos\theta\)	\(\tan\theta\)

The operation was reflect. The signs are the coordinates of the image point, not a second family of functions.

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4. Cofunctions — swap the two legs

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The complement \(\pi/2-\theta\) swaps opposite and adjacent. On the unit circle the orange and green legs trade roles: the run of one angle is the rise of the other.

\[ \boxed{\sin(\pi/2-\theta)=\cos\theta,\qquad \cos(\pi/2-\theta)=\sin\theta} \]

\[ \boxed{\tan(\pi/2-\theta)=\cot\theta,\qquad \cot(\pi/2-\theta)=\tan\theta} \]

\[ \boxed{\sec(\pi/2-\theta)=\csc\theta,\qquad \csc(\pi/2-\theta)=\sec\theta} \]

Cofunction names are literal: co-sine is the sine of the complement. Swapping the two legs is the whole operation.

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5. Period and coterminal angles — same point \(P\)

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Adding a full turn \(2\pi k\) does not move \(P\). The six segments are the same lengths, so sine, cosine, and their reciprocals repeat every \(2\pi\). Tangent and cotangent are slopes of the same ray; a half-turn \(\pi\) sends the ray onto itself with both legs flipped, so the ratio repeats every \(\pi\).

No new figure: the six-panel graphs in §11 make the periods visible as the first repeat of each wave.

\[ \boxed{\sin(\theta+2\pi k)=\sin\theta,\quad \cos(\theta+2\pi k)=\cos\theta} \]

\[ \boxed{\sec(\theta+2\pi k)=\sec\theta,\quad \csc(\theta+2\pi k)=\csc\theta} \]

\[ \boxed{\tan(\theta+\pi k)=\tan\theta,\quad \cot(\theta+\pi k)=\cot\theta} \]

Coterminal angles land on the same point \(P\). That is the operation: add a full turn (or a half-turn for tan and cot) and nothing on the circle moves.

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6. Angle addition and subtraction — compose two rotations

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Turning by \(A\) and then by \(B\) is one rotation by \(A+B\). The new height at \(P_{A+B}\) is not a height you already drew — it mixes the two pairs: each old sine and cosine appears in a product, and those products add.

\[ \boxed{\sin(A+B)=\sin A\cos B+\cos A\sin B} \]

\[ \boxed{\sin(A-B)=\sin A\cos B-\cos A\sin B} \]

\[ \boxed{\cos(A+B)=\cos A\cos B-\sin A\sin B} \]

\[ \boxed{\cos(A-B)=\cos A\cos B+\sin A\sin B} \]

\[ \boxed{\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}} \]

\[ \boxed{\tan(A-B)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}} \]

Compose two rotations; the boxed lines are the coordinates of the composite point. §14 is the same fact written as \(e^{iA}\,e^{iB}=e^{i(A+B)}\).

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7. Double angle, difference of squares, power-reduction

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Set \(B=A=\theta\) in §6. Squaring and subtracting the two legs is \(-\cos 2\theta\), not a new function. Doubling the angle is the same mix, with both pairs taken from one ray.

\[ \boxed{\sin 2\theta=2\sin\theta\cos\theta} \]

\[ \boxed{\cos 2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta} \]

\[ \boxed{\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}} \]

\[ \boxed{\sin^2\theta-\cos^2\theta=-\cos 2\theta} \]

Power-reduction is the same algebra solved for the squares — the Fourier / average form, not a third identity family:

\[ \boxed{\sin^2\theta=(1-\cos 2\theta)/2} \]

\[ \boxed{\cos^2\theta=(1+\cos 2\theta)/2} \]

\[ \boxed{\tan^2\theta=(1-\cos 2\theta)/(1+\cos 2\theta)} \]

The operation was square and subtract (or rearrange). Half the frequency appears because the product of a wave with itself carries a double-angle term.

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8. Half-angle and Weierstrass — square-root undoes the square

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Solving the double-angle and power-reduction formulas of §7 for the half-argument is a square root. The sign follows the quadrant of \(\theta/2\). The figure is the same \(\theta\) versus \(2\theta\) picture — no tenth circle.

\[ \boxed{\sin(\theta/2)=\pm\sqrt{(1-\cos\theta)/2}} \]

\[ \boxed{\cos(\theta/2)=\pm\sqrt{(1+\cos\theta)/2}} \]

\[ \boxed{\tan(\theta/2)=\pm\sqrt{(1-\cos\theta)/(1+\cos\theta)}} \]

The square-root form of tangent can pick the wrong sign. These two algebraic forms stay on the correct branch whenever they are defined:

\[ \boxed{\tan(\theta/2)=\dfrac{\sin\theta}{1+\cos\theta}=\dfrac{1-\cos\theta}{\sin\theta}} \]

Weierstrass substitution — one letter \(t\) for that half-angle tangent:

\[ \boxed{t=\tan(\theta/2)\quad\Rightarrow\quad \sin\theta=\dfrac{2t}{1+t^2},\quad \cos\theta=\dfrac{1-t^2}{1+t^2},\quad \tan\theta=\dfrac{2t}{1-t^2}} \]

This is the substitution used to turn a rational function of \(\sin\theta\) and \(\cos\theta\) into an ordinary rational function of \(t\) — the standard integral trick. It is the same half-angle geometry: \(t\) is the purple tangent of the half-ray, and the three formulas are §7 read from \(\theta/2\) up to \(\theta\).

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9. Product-to-sum and sum-to-product — multiply two heights

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The product of two sinusoids is not a new shape. It is a sum of two waves whose frequencies are the sum and the difference. On the figure, the pointwise product of the two heights is redrawn as that pair of waves. The \(a,b,R\) inset in the same figure is for §10.

\[ \boxed{\sin A\sin B=[\cos(A-B)-\cos(A+B)]/2} \]

\[ \boxed{\cos A\cos B=[\cos(A+B)+\cos(A-B)]/2} \]

\[ \boxed{\sin A\cos B=[\sin(A+B)+\sin(A-B)]/2} \]

\[ \boxed{\sin C+\sin D=2\sin((C+D)/2)\cos((C-D)/2)} \]

\[ \boxed{\sin C-\sin D=2\cos((C+D)/2)\sin((C-D)/2)} \]

\[ \boxed{\cos C+\cos D=2\cos((C+D)/2)\cos((C-D)/2)} \]

\[ \boxed{\cos C-\cos D=-2\sin((C+D)/2)\sin((C-D)/2)} \]

Multiply two heights; the boxed lines are the sum-and-difference rewrite. The reverse direction (sum-to-product) is the same operation read the other way — useful when a sum should become a factored amplitude.

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10. Harmonic addition — scale, add, one shifted wave

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A scaled sine plus a scaled cosine is not a new shape. It is one sine with amplitude \(R=\sqrt{a^2+b^2}\) and a phase shift. The right triangle of legs \(a,b\) and hypotenuse \(R\) is the inset on the §9 wave figure — that triangle is the operation.

\[ \boxed{a\sin\theta+b\cos\theta=R\sin(\theta+\varphi),\quad R=\sqrt{a^2+b^2},\quad \cos\varphi=a/R,\quad \sin\varphi=b/R} \]

The same vector \((a,b)\) can be written as a shifted cosine: \(a\sin\theta+b\cos\theta=R\cos(\theta-\alpha)\) for a phase \(\alpha\) read from the same inset. Scale, add; one shifted wave remains.

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11. Six function graphs

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The six functions of §1, plotted against \(\theta\) in the same colors as the unit-circle segments. Period, range, and vertical asymptotes are the graph’s way of saying when a tangent is missed — the same undefined cases as on the diagram.

Function	Period	Range	Vertical asymptotes
\(\sin\theta\)	\(2\pi\)	\([-1,1]\)	none
\(\cos\theta\)	\(2\pi\)	\([-1,1]\)	none
\(\tan\theta\)	\(\pi\)	\(\mathbb{R}\)	\(\theta=\pi/2+k\pi\)
\(\cot\theta\)	\(\pi\)	\(\mathbb{R}\)	\(\theta=k\pi\)
\(\sec\theta\)	\(2\pi\)	\((-\infty,-1]\cup[1,\infty)\)	\(\theta=\pi/2+k\pi\)
\(\csc\theta\)	\(2\pi\)	\((-\infty,-1]\cup[1,\infty)\)	\(\theta=k\pi\)

Near \(\theta=0\) the green height and the arc look interchangeable. Those small-angle approximations live in §14, as the first terms of \(e^{i\theta}\). This section is the global picture: period, range, and which rays never meet a tangent.

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12. Special angles and CAST

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A handful of first-quadrant angles have exact ratios. Reciprocals are \(1/x\) of those values — no second memorization table. Signs through a full turn are CAST: which functions stay positive in each quadrant.

\(\theta\)	\(0^\circ\) (\(0\))	\(30^\circ\) (\(\pi/6\))	\(45^\circ\) (\(\pi/4\))	\(60^\circ\) (\(\pi/3\))	\(90^\circ\) (\(\pi/2\))
\(\sin\theta\)	\(0\)	\(1/2\)	\(\sqrt{2}/2\)	\(\sqrt{3}/2\)	\(1\)
\(\cos\theta\)	\(1\)	\(\sqrt{3}/2\)	\(\sqrt{2}/2\)	\(1/2\)	\(0\)
\(\tan\theta\)	\(0\)	\(1/\sqrt{3}\)	\(1\)	\(\sqrt{3}\)	undefined
\(\csc\theta\)	undefined	\(1/(1/2)=2\)	\(1/(\sqrt{2}/2)=\sqrt{2}\)	\(1/(\sqrt{3}/2)=2/\sqrt{3}\)	\(1/1=1\)
\(\sec\theta\)	\(1/1=1\)	\(1/(\sqrt{3}/2)=2/\sqrt{3}\)	\(1/(\sqrt{2}/2)=\sqrt{2}\)	\(1/(1/2)=2\)	undefined
\(\cot\theta\)	undefined	\(1/(1/\sqrt{3})=\sqrt{3}\)	\(1/1=1\)	\(1/\sqrt{3}\)	\(0\)

CAST / quadrant signs through one turn — signs only, not a grid of every \(15^\circ\):

Quadrant	Interval	\(\sin,\csc\)	\(\cos,\sec\)	\(\tan,\cot\)	Memory
I	\((0,\pi/2)\)	\(+\)	\(+\)	\(+\)	All
II	\((\pi/2,\pi)\)	\(+\)	\(-\)	\(-\)	Sin
III	\((\pi,3\pi/2)\)	\(-\)	\(-\)	\(+\)	Tan
IV	\((3\pi/2,2\pi)\)	\(-\)	\(+\)	\(-\)	Cos

Exact values are first-quadrant lengths. CAST paints the sign after you reduce by §3.

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13. Law of Sines and Cosines — the same ratios, triangle no longer right

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Drop an altitude in a scalene triangle and the right-triangle ratios of the unit-circle diagram reappear on each piece. The Laws package that construction so you do not redraw the altitude every time. Side \(a\) sits opposite angle \(A\), and \(R\) here is the circumradius — not the harmonic-addition amplitude of §10.

\[ \boxed{a/\sin A=b/\sin B=c/\sin C=2R} \]

\[ \boxed{c^2=a^2+b^2-2ab\cos C} \]

\[ \boxed{\text{Area}=\tfrac12 ab\sin C} \]

The cosine law is cyclic: \(a^2=b^2+c^2-2bc\cos A\) and \(b^2=a^2+c^2-2ac\cos B\). When \(C=90^\circ\), \(\cos C=0\) and the Law of Cosines is §2. The same two legs, the same square-and-add, now written for a triangle that is no longer required to be right.

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14. Euler — the point \((\cos\theta, \sin\theta)\) is \(e^{i\theta}\)

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The unit-circle point \(P=(\cos\theta,\sin\theta)\) is one complex number. Multiplying by \(e^{i\alpha}\) is rotation by \(\alpha\) — that is why the addition formulas of §6 work. The green height and the orange run are the imaginary and real parts of the same exponential.

\[ \boxed{e^{i\theta}=\cos\theta+i\sin\theta} \]

\[ \boxed{(e^{i\theta})^n=e^{in\theta}=\cos(n\theta)+i\sin(n\theta)} \]

\[ \boxed{\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}} \]

De Moivre is the second line: raising a unit complex number to the \(n\)th power walks \(n\) equal steps around the circle. The \(2\times 2\) matrix is that same rotation acting on a real plane vector.

One-liner exponential definitions: \(\cos\theta=(e^{i\theta}+e^{-i\theta})/2\) and \(\sin\theta=(e^{i\theta}-e^{-i\theta})/(2i)\).

Near \(\theta=0\) (radians) the green height and the arc agree, and the orange run sits just below \(1\). Those are the first Taylor terms of \(e^{i\theta}\):

\[ \boxed{\sin\theta\approx\theta,\qquad \tan\theta\approx\theta,\qquad \cos\theta\approx 1-\theta^2/2} \]

Quantum amplitudes and signal-processing phasors reuse this picture: a state or a tone is a point on the same circle, and a phase shift is multiplication by \(e^{i\alpha}\). The handbook stops at the geometry; the later series articles pick up the physics.

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15. How to use this handbook

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Ask which operation am I doing?, then jump:

1. Square the legs and add — §2. Two lengths make hypotenuse \(1\).
2. Reflect the ray — §3. Odd/even and the allied-angle table.
3. Swap the two legs — §4. Complements.
4. Add a full turn — §5. Same point \(P\).
5. Compose two rotations — §6. Plus between the sine products.
6. Square and subtract / double — §7. Power-reduction is the same algebra.
7. Take a square root — §8. Weierstrass \(t=\tan(\theta/2)\) for integrals.
8. Multiply two heights — §9.
9. Scale and add two waves — §10. One shifted sine; see the \(a,b,R\) inset.
10. Read the graphs — §11. Period, range, asymptotes.
11. Need an exact value or a sign — §12.
12. The triangle is no longer right — §13.
13. One complex point, or a small angle — §14.

Triple-angle is not a third family. Expand \(\sin(2\theta+\theta)\) from §6 when you need it: \(\sin 3\theta=\sin(2\theta+\theta)=\sin 2\theta\cos\theta+\cos 2\theta\sin\theta\), then substitute §7. Do not memorize a separate triple-angle list.

Once the operation is named, the boxed line is a measurement you can see. The six colored segments stay the same lengths they were in the first article; this handbook is what you are allowed to do to them.

Hashtags

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#Trigonometry #TrigIdentities #UnitCircle #PythagoreanIdentity #AngleAddition #EulerFormula #LawOfSines #LawOfCosines #MathForML #Visualization #AppliedMathematics #QuantumMath