Trigonometric Functions

⚡ Key Takeaways

Trigonometric functions (sine, cosine, tangent, and their reciprocals) relate the angles of a triangle to ratios of its side lengths, and more generally express periodic oscillation via the unit circle. Key result: for any real angle θ\theta, the point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) lies on the unit circle, giving sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Where it applies: geometry, waves, oscillations, navigation, signal processing, and all of physics involving rotation or periodicity. Who/when: developed from Hipparchus (~150 BCE) through Indian, Islamic, and European mathematicians into the modern functional form by Euler (18th century).

💡 The Core Insight

A single rotating point on a circle of radius 1 encodes both sine and cosine simultaneously: its horizontal coordinate is cos⁡θ\cos\theta and its vertical coordinate is sin⁡θ\sin\theta. As the angle sweeps round, these coordinates trace out smooth waves. Every trigonometric identity is, at heart, a statement about this rotating point and the geometry of the circle.

Formal Statement

DefinitionRight-Triangle Definitions

For an acute angle θ\theta in a right triangle, label the side opposite θ\theta as OO, the side adjacent to θ\theta as AA, and the hypotenuse as HH (the side opposite the right angle). Then:

1
sin⁡θ=OH,cos⁡θ=AH,tan⁡θ=OA\quad \sin\theta = \frac{O}{H}, \qquad \cos\theta = \frac{A}{H}, \qquad \tan\theta = \frac{O}{A}

Derivation hint: These are definitions; the mnemonic SOH-CAH-TOA encodes them. Ratios are independent of triangle size by similarity. The three reciprocal functions are:

2
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\quad \csc\theta = \frac{1}{\sin\theta}, \qquad \sec\theta = \frac{1}{\cos\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Derivation hint: Cosecant, secant, cotangent are defined as reciprocals of sine, cosine, tangent respectively. Here O,A,H>0O, A, H > 0 are side lengths and 0<θ<90°0 < \theta < 90° (or 0<θ<π/20 < \theta < \pi/2 radians).

DefinitionUnit-Circle Definitions

To extend the functions to all real angles, place a point PP on the unit circle x2+y2=1x^2 + y^2 = 1 at angle θ\theta measured counterclockwise from the positive xx-axis. Then:

3
cos⁡θ=xP,sin⁡θ=yP\quad \cos\theta = x_P, \qquad \sin\theta = y_P

Derivation hint: For a unit circle, H=1H = 1, so sin⁡θ=O/1=y\sin\theta = O/1 = y and cos⁡θ=A/1=x\cos\theta = A/1 = x, generalised by allowing x,yx, y to be negative. Angles are best measured in radians, where a full revolution is 2π2\pi:

4
θrad=θdeg×π180\quad \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Derivation hint: One full circle subtends arc length 2πr2\pi r; defining radian measure as arc/radius gives 2π2\pi rad =360°= 360°.

TheoremPythagorean Identity

For every real θ\theta:

5
sin⁡2θ+cos⁡2θ=1\quad \sin^2\theta + \cos^2\theta = 1

Derivation hint: Apply the equation of the unit circle x2+y2=1x^2 + y^2 = 1 with x=cos⁡θx = \cos\theta, y=sin⁡θy = \sin\theta. Dividing by cos⁡2θ\cos^2\theta and by sin⁡2θ\sin^2\theta yields two companions:

6
1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ\quad 1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \csc^2\theta

Derivation hint: Divide [5] through by cos⁡2θ\cos^2\theta for the first; by sin⁡2θ\sin^2\theta for the second; use definitions [2].

TheoremAngle Sum Formulas

For all real α,β\alpha, \beta:

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sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\quad \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta

Derivation hint: Rotate a unit vector by α\alpha then β\beta; equate components, or use the rotation matrix product / Euler's formula.

8
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\quad \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

Derivation hint: Same rotation argument; note the sign reversal — "±\pm" on the left gives "∓\mp" on the right.

9
tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\quad \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

Derivation hint: Divide [7] by [8] and simplify by dividing numerator and denominator by cos⁡αcos⁡β\cos\alpha\cos\beta.

Why It Holds / Derivation

ProofPythagorean Identity from the Unit Circle

1. Define P=(cos⁡θ,sin⁡θ)P = (\cos\theta, \sin\theta) as the point on the unit circle at angle θ\theta, by [3]. 2. By construction PP lies on the circle x2+y2=1x^2 + y^2 = 1. 3. Substitute x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta directly into the circle equation. 4. This gives cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1, establishing [5]. 5. Because PP exists for every real θ\theta (the point simply wraps around the circle), the identity holds universally — not merely for acute angles. ∎

∎
ProofCosine Sum Formula via Distance

1. Place two points on the unit circle: A=(cos⁡α,sin⁡α)A = (\cos\alpha, \sin\alpha) and B=(cos⁡β,sin⁡β)B = (\cos\beta, \sin\beta). 2. The chord length squared is ∣AB∣2=(cos⁡α−cos⁡β)2+(sin⁡α−sin⁡β)2|AB|^2 = (\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2. 3. Expanding and using [5] twice: ∣AB∣2=2−2(cos⁡αcos⁡β+sin⁡αsin⁡β)|AB|^2 = 2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta). 4. Now rotate both points so BB sits at angle 00 and AA at angle (α−β)(\alpha - \beta). Rotation preserves distance, so ∣AB∣2=(cos⁡(α−β)−1)2+sin⁡2(α−β)|AB|^2 = (\cos(\alpha-\beta) - 1)^2 + \sin^2(\alpha-\beta). 5. Expanding step 4: ∣AB∣2=2−2cos⁡(α−β)|AB|^2 = 2 - 2\cos(\alpha - \beta). 6. Equate steps 3 and 5: cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta, which is [8] with the "−-" case. ∎

∎
mathematician · Leonhard Euler | 1707–1783

Euler transformed trigonometry from a study of triangles into a branch of analysis. He standardised the modern notation (sin⁡\sin, cos⁡\cos, tan⁡\tan), treated the functions as functions of a real variable rather than chords, and discovered the celebrated identity eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta, which unifies trigonometric and exponential functions. From this single formula, every angle-sum identity falls out as a consequence of the laws of exponents.

Building Intuition

🧠 Intuition

Imagine a clock hand of length 1 pinned at the origin. As it spins counterclockwise, its tip has a shadow on the floor (the xx-axis) and a shadow on the wall (the yy-axis). The floor-shadow is cos⁡θ\cos\theta; the wall-shadow is sin⁡θ\sin\theta. When the hand points right, the floor-shadow is full (cos = 1) and the wall-shadow vanishes (sin = 0). A quarter-turn later they swap. This is why sine and cosine are the same wave shifted by a quarter cycle: cos⁡θ=sin⁡(θ+π/2)\cos\theta = \sin(\theta + \pi/2). The tangent measures the slope of the hand — it shoots to infinity when the hand points straight up because a vertical line has undefined slope.

Figure
The unit circle generating sine as height and cosine as horizontal projection

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P=(cos θ, sin θ) sin θ cos θ θ sin as a wave

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Worked Examples

ExampleFinding a Side with Tangent

Given: A ladder leans against a wall at an angle of θ=65°\theta = 65° to the ground. Its base is 2.0 m2.0\ \text{m} from the wall. How high up the wall does it reach?

Steps: 1. The adjacent side (base distance) is A=2.0 mA = 2.0\ \text{m}; the opposite side (height hh) is unknown. 2. Use tan⁡θ=O/A\tan\theta = O/A from [1]: tan⁡65°=h/2.0\tan 65° = h / 2.0. 3. tan⁡65°≈2.1445\tan 65° \approx 2.1445. 4. Solve: h=2.0×2.1445=4.289 mh = 2.0 \times 2.1445 = 4.289\ \text{m}.

h≈4.3 m\boxed{h \approx 4.3\ \text{m}}
ExampleEvaluating an Exact Angle-Sum

Given: Find the exact value of sin⁡75°\sin 75°.

Steps: 1. Write 75°=45°+30°75° = 45° + 30° and apply [7] with the "++" case. 2. sin⁡(45°+30°)=sin⁡45°cos⁡30°+cos⁡45°sin⁡30°\sin(45° + 30°) = \sin 45°\cos 30° + \cos 45°\sin 30°. 3. Substitute exact values: sin⁡45°=cos⁡45°=22\sin 45° = \cos 45° = \tfrac{\sqrt 2}{2}, cos⁡30°=32\cos 30° = \tfrac{\sqrt 3}{2}, sin⁡30°=12\sin 30° = \tfrac{1}{2}. 4. =22⋅32+22⋅12=64+24= \tfrac{\sqrt 2}{2}\cdot\tfrac{\sqrt 3}{2} + \tfrac{\sqrt 2}{2}\cdot\tfrac{1}{2} = \tfrac{\sqrt 6}{4} + \tfrac{\sqrt 2}{4}.

sin⁡75°=6+24≈0.9659\boxed{\sin 75° = \frac{\sqrt 6 + \sqrt 2}{4} \approx 0.9659}

Common Misconceptions

⚠️ Common mistake

sin⁡2θ\sin^2\theta does not mean sin⁡(sin⁡θ)\sin(\sin\theta). The notation sin⁡2θ\sin^2\theta is shorthand for (sin⁡θ)2(\sin\theta)^2. However, sin⁡−1θ\sin^{-1}\theta does not mean 1/sin⁡θ1/\sin\theta — it denotes the inverse function (arcsine). The reciprocal is written csc⁡θ\csc\theta or (sin⁡θ)−1(\sin\theta)^{-1}. This inconsistency in the "−1^{-1}" superscript is a frequent source of error.

⚠️ Common mistake

Always check whether your calculator is in degrees or radians. Computing sin⁡30\sin 30 in radian mode gives ≈−0.988\approx -0.988 (the sine of 30 radians), not the intended 0.50.5. Calculus formulas such as ddθsin⁡θ=cos⁡θ\tfrac{d}{d\theta}\sin\theta = \cos\theta are only valid in radians.

Counterexample

Sine is not linear: sin⁡(α+β)≠sin⁡α+sin⁡β\sin(\alpha + \beta) \neq \sin\alpha + \sin\beta in general. Take α=β=90°\alpha = \beta = 90°. Then sin⁡(180°)=0\sin(180°) = 0, but sin⁡90°+sin⁡90°=1+1=2\sin 90° + \sin 90° = 1 + 1 = 2. The correct expansion requires the angle-sum formula [7].

Connections

connection

Trigonometric functions sit at the crossroads of many fields. Through Euler's Formula they merge with the complex Exponential Function, so that cos⁡θ\cos\theta and sin⁡θ\sin\theta become the real and imaginary parts of eiθe^{i\theta}. Their derivatives and integrals are central to Calculus, where sin⁡\sin and cos⁡\cos form a closed cycle under differentiation. The Law of Sines and Law of Cosines extend the right-triangle ratios to any triangle. In analysis, sums of sines and cosines build the Fourier Series, decomposing arbitrary periodic signals into pure tones.

Real-World Applications

🔬 Where this appears

Historical Development

⏳ History
- ~150 BCE — Hipparchus of Nicaea compiles the first known table of chords, a precursor to sine, for astronomical prediction.
- ~150 CE — Ptolemy refines chord tables in the Almagest and derives chord-addition formulas equivalent to the angle-sum identities.
- ~500 CE — Aryabhata (India) introduces the half-chord (jya), the direct ancestor of the modern sine.
- ~830 CE — Al-Khwārizmī & later Al-Battānī (Islamic world) develop sine and tangent tables and spherical trigonometry for astronomy and qibla direction.
- ~1000 CE — Abū al-Wafā states identities for the six functions and improves table accuracy.
- 1464 — Regiomontanus writes De triangulis omnimodis, the first European treatise treating trigonometry as an independent discipline.
- 1614 — John Napier invents logarithms, vastly simplifying trigonometric computation.
- 1748 — Leonhard Euler publishes Introductio in analysin infinitorum, defining the functions analytically and proving eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta.
- 1822 — Joseph Fourier shows that arbitrary functions can be expressed as infinite sums of sines and cosines, founding harmonic analysis.

Subtleties & Limits

open-problem

At the introductory level the trig functions seem fully tamed, but subtleties remain. The inverse functions are multivalued — sin⁡x=0.5\sin x = 0.5 has infinitely many solutions — so principal-value branches must be chosen carefully, a recurring difficulty in solving equations and in complex analysis. Extending the functions to complex arguments makes sine and cosine unbounded (e.g. cos⁡(i)≈1.543>1\cos(i) \approx 1.543 > 1), defying naive intuition. Deep number-theoretic questions persist too: the exact nature of values like sin⁡(1°)\sin(1°) involves algebraic numbers, and which rational multiples of π\pi give rational sine values is settled (Niven's theorem) but related transcendence questions about sums of such values remain delicate research topics.

See Also

The Unit Circle · Law of Sines · Law of Cosines · Hyperbolic Functions · Periodic Functions · Simple Harmonic Motion