Trigonometric Properties

⚡ Key Takeaways

Trigonometric properties are the algebraic relationships (identities) connecting the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. The key result: these functions are bound together by the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ and an entire web of derived relations that hold for every angle. They apply wherever angles, rotation, waves, or periodic phenomena appear — from surveying to signal processing. Systematised by Greek, Indian, and Islamic mathematicians between roughly 150 BCE and 1000 CE.

💡 The Core Insight

A point moving on the unit circle has coordinates $(\cos\theta,\sin\theta)$ . Because that point always lies a distance 1 from the centre, its coordinates must satisfy $x^2+y^2=1$ . Every trigonometric property is a consequence of this fact plus the geometry of rotating that point. Master the unit circle, and you do not need to memorise identities — you can rebuild them.

DefinitionThe Six Functions on the Unit Circle

For an angle $\theta$ measured counterclockwise from the positive $x$ -axis, let $P=(x,y)$ be the corresponding point on the unit circle. The six functions are defined as follows.

1

\quad \cos\theta = x,\qquad \sin\theta = y

Derivation hint: definitions — the coordinates of the unit-circle point ARE cosine and sine.

2

\quad \tan\theta=\frac{\sin\theta}{\cos\theta},\qquad \cot\theta=\frac{\cos\theta}{\sin\theta}

Derivation hint: tangent is the slope $y/x$ of the radius to $P$ ; cotangent is its reciprocal.

3

\quad \sec\theta=\frac{1}{\cos\theta},\qquad \csc\theta=\frac{1}{\sin\theta}

Derivation hint: definitions of the reciprocal functions; valid where denominators are nonzero. Here $\theta$ is the angle (in radians or degrees), $x$ and $y$ are the horizontal and vertical coordinates of $P$ , and all quantities are dimensionless ratios.

DefinitionRight-Triangle Form

For an acute angle $\theta$ in a right triangle with hypotenuse $h$ , opposite side $o$ , and adjacent side $a$ :

4

\quad \sin\theta=\frac{o}{h},\quad \cos\theta=\frac{a}{h},\quad \tan\theta=\frac{o}{a}

Derivation hint: the unit-circle definitions scaled by the hypotenuse $h$ ; equivalent for acute angles.

TheoremThe Fundamental Identities

For all angles $\theta$ (where defined), the following hold.

Pythagorean identities:

5

\quad \sin^2\theta+\cos^2\theta=1

Derivation hint: apply $x^2+y^2=1$ with $x=\cos\theta,\ y=\sin\theta$ .

6

\quad 1+\tan^2\theta=\sec^2\theta

Derivation hint: divide identity [5] through by $\cos^2\theta$ .

7

\quad 1+\cot^2\theta=\csc^2\theta

Derivation hint: divide identity [5] through by $\sin^2\theta$ .

Angle-sum identities:

8

\quad \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta

Derivation hint: rotate the unit-circle point through $\alpha$ then $\beta$ ; or use the rotation matrix product.

9

\quad \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta

Derivation hint: same rotation construction; signs reverse relative to the sine formula.

Double-angle identities:

10

\quad \sin 2\theta=2\sin\theta\cos\theta,\qquad \cos 2\theta=\cos^2\theta-\sin^2\theta

Derivation hint: set $\alpha=\beta=\theta$ in [8] and [9].

Even/odd (symmetry) and periodicity:

11

\quad \cos(-\theta)=\cos\theta,\quad \sin(-\theta)=-\sin\theta,\quad \sin(\theta+2\pi)=\sin\theta

Derivation hint: reflecting $P$ across the $x$ -axis negates $y$ but keeps $x$ ; a full turn returns to start.

ProofThe Pythagorean Identity from the Unit Circle

1. Place an angle $\theta$ in standard position; let $P=(x,y)$ be where its ray crosses the unit circle. 2. By the distance formula, the distance from the origin $O=(0,0)$ to $P$ is $\sqrt{x^2+y^2}$ . 3. By definition the radius of the unit circle is $1$ , so $\sqrt{x^2+y^2}=1$ , hence $x^2+y^2=1$ . 4. Substitute the definitions [1], $x=\cos\theta$ and $y=\sin\theta$ , to obtain $\cos^2\theta+\sin^2\theta=1$ , which is identity [5]. $\blacksquare$

∎

ProofDeriving the Tangent Pythagorean Identity

1. Start from identity [5]: $\sin^2\theta+\cos^2\theta=1$ . 2. Assume $\cos\theta\neq0$ and divide every term by $\cos^2\theta$ :

12

\quad \frac{\sin^2\theta}{\cos^2\theta}+\frac{\cos^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}

Derivation hint: legitimate division because $\cos^2\theta\neq0$ . 3. Recognise $\sin\theta/\cos\theta=\tan\theta$ (from [2]) and $1/\cos\theta=\sec\theta$ (from [3]). 4. This yields $\tan^2\theta+1=\sec^2\theta$ , which is identity [6]. $\blacksquare$

∎

mathematician · Hipparchus of Nicaea | c. 190 – c. 120 BCE

Often called the founder of trigonometry, Hipparchus compiled the first known table of chords — a precursor to the sine table — to solve problems in astronomy. He needed to relate the central angle of a circle to the length of the chord it subtends, effectively tabulating $2\sin(\theta/2)$ . His work on the lengths of chords made the systematic study of triangle-angle relationships possible and underpinned Ptolemy's later Almagest.

🧠 Intuition

Picture a clock hand of length 1 sweeping around. Its shadow on the floor (horizontal projection) is $\cos\theta$ ; its shadow on the wall (vertical projection) is $\sin\theta$ . As the hand turns, these two shadows trade size: when one is long the other is short, and they do so in exactly the way that keeps $\sin^2+\cos^2$ pinned at 1. The Pythagorean identity is just the statement that the hand never changes length. Tangent is the slope of the hand — steep near vertical (blowing up to infinity at $90°$ ), flat near horizontal.

Figure

The unit circle: cosine and sine as projections of a rotating radius

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ExampleSimplifying with the Pythagorean Identity

Given: Simplify $\dfrac{\sin^2\theta}{1-\cos\theta}$ .

Steps: 1. From identity [5], $\sin^2\theta=1-\cos^2\theta$ . 2. Factor the numerator as a difference of squares: $1-\cos^2\theta=(1-\cos\theta)(1+\cos\theta)$ . 3. Substitute:

13

\quad \frac{\sin^2\theta}{1-\cos\theta}=\frac{(1-\cos\theta)(1+\cos\theta)}{1-\cos\theta}

Derivation hint: factor numerator, then cancel the common factor (valid for $\cos\theta\neq1$ ). 4. Cancel $(1-\cos\theta)$ .

\boxed{\dfrac{\sin^2\theta}{1-\cos\theta}=1+\cos\theta\quad(\cos\theta\neq1)}

ExampleComputing an Exact Value with the Sum Formula

Given: Find the exact value of $\cos 75°$ (dimensionless).

Steps: 1. Write $75°=45°+30°$ — two angles with known exact values. 2. Apply identity [9] with the $+$ case:

14

\quad \cos(45°+30°)=\cos45°\cos30°-\sin45°\sin30°

Derivation hint: substitute $\alpha=45°,\ \beta=30°$ into the cosine sum formula. 3. Insert exact values $\cos45°=\sin45°=\tfrac{\sqrt2}{2}$ , $\cos30°=\tfrac{\sqrt3}{2}$ , $\sin30°=\tfrac12$ :

15

\quad \cos75°=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac12=\frac{\sqrt6-\sqrt2}{4}

Derivation hint: multiply fractions and combine over the common denominator 4.

\boxed{\cos 75°=\dfrac{\sqrt6-\sqrt2}{4}\approx 0.2588}

⚠️ Common mistake

$\sin^2\theta$ does NOT mean $\sin(\sin\theta)$ . The notation $\sin^2\theta$ is shorthand for $(\sin\theta)^2$ . Worse, $\sin^{-1}\theta$ means the inverse function $\arcsin\theta$ , not $1/\sin\theta=\csc\theta$ . The exponent notation is genuinely inconsistent, and you must read it by convention: positive integer powers multiply, but $-1$ signals inversion.

⚠️ Common mistake

The angle-sum identities do not distribute. It is a common and serious error to write $\sin(\alpha+\beta)=\sin\alpha+\sin\beta$ . This is almost always false. The correct expansion is identity [8], which mixes both functions of both angles.

Counterexample

Test $\sin(\alpha+\beta)=\sin\alpha+\sin\beta$ with $\alpha=\beta=90°$ . The left side is $\sin180°=0$ . The right side is $\sin90°+\sin90°=1+1=2$ . Since $0\neq2$ , the "distributed" version is false. Only the genuine identity [8] gives $\sin180°=\sin90°\cos90°+\cos90°\sin90°=0+0=0$ . ✓

connection

Trigonometric properties tie together a remarkable swath of mathematics. The Pythagorean identity is literally the Pythagorean Theorem in disguise. The angle-sum formulas are encoded compactly by Euler's Formula, $e^{i\theta}=\cos\theta+i\sin\theta$ , which converts trigonometric identities into ordinary exponent laws. The periodicity of these functions makes them the building blocks of Fourier Series, and their derivatives ( $\frac{d}{dx}\sin x=\cos x$ ) link them to Differential Calculus. The Law of Cosines generalises the Pythagorean identity to non-right triangles.

🔬 Where this appears

⏳ History

- c. 150 BCE — Hipparchus compiles the first table of chords, founding systematic trigonometry for astronomy.

- c. 150 CE — Ptolemy publishes the Almagest, including "Ptolemy's theorem," from which the angle-sum identities can be derived geometrically.

- c. 400–500 CE — Aryabhata (India) introduces the half-chord (jya), essentially the modern sine, and tabulates its values.

- c. 628 CE — Brahmagupta develops interpolation formulas for sine tables.

- c. 980 CE — Abu al-Wafa (Baghdad) introduces the tangent and secant functions and proves the sine addition formula explicitly.

- c. 1000 CE — al-Biruni applies trigonometry to compute the Earth's radius and refines spherical trigonometry.

- 1464 CE — Regiomontanus writes De Triangulis, the first European text treating trigonometry as an independent subject.

- 1748 CE — Leonhard Euler unifies trigonometry with the exponential function via

e^{i\theta}=\cos\theta+i\sin\theta

, reframing identities as algebra.

open-problem

The identities hold "where defined," and the exceptions matter. Functions like $\tan$ , $\sec$ , $\csc$ , and $\cot$ have domain restrictions — they are undefined wherever a denominator vanishes (e.g., $\tan\theta$ at $\theta=90°$ ), and an identity manipulation that cancels a factor like $(1-\cos\theta)$ silently excludes points where that factor is zero. A subtler open-ended question concerns generalisation: the functions extend to complex arguments (where $\cos z$ and $\sin z$ are unbounded), to hyperbolic analogues ( $\cosh$ , $\sinh$ with $\cosh^2-\sinh^2=1$ ), and to q-deformed and spherical versions — and tracking which classical identities survive each generalisation remains a rich theme in higher mathematics.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also