Trigonometric Functions – Comprehensive Bullet-Point Notes

Introduction

  • “Trigonometry” comes from Greek words “trigon” (triangle) + “metron” (measure)

  • Originated to solve triangle problems; used today in seismology, electronics, atomic physics, oceanography, music analysis, engineering, navigation, surveying, etc.

  • Earlier classes: acute–angle ratios, identities, heights & distances. Chapter focus: extend ratios to functions valid for any real angle (radian measure), explore properties

Angle: Concept & Terminology

  • Angle = amount of rotation of a ray about its initial point

    • Initial side → original ray

    • Terminal side → final ray after rotation

    • Vertex → common initial point

  • Direction

    • Anticlockwise rotation ⇒ positive angle

    • Clockwise rotation ⇒ negative angle

  • Quadrantal angles: integral multiples of \frac{\pi}{2} (0, \frac{\pi}{2}, \pi, \tfrac{3\pi}{2}, 2\pi …)

Units of Angle Measure

  • Revolution

    • 1 complete revolution = 2\pi rad = 360^\circ

  • Degree (°)

    • 1^\circ = \frac{1}{360} revolution

    • Sexagesimal subdivisions: 1^\circ = 60' ; 1' = 60''

  • Radian (rad)

    • In unit circle (radius = 1), angle subtended by arc length 1 is 1 rad

    • General circle: angle subtended by arc length l on radius r given by \theta = \frac{l}{r} ⇒ l = r\theta

Relation: Real Numbers ↔ Radians

  • Each real number plotted on tangent line through point A(1,0) of unit circle corresponds to a unique arc length ⇒ unique radian measure; radian measures and real numbers are bijective

Degree–Radian Conversion

  • Fundamental: 2\pi\text{ rad}=360^{\circ} ⇒ \pi\text{ rad}=180^{\circ}

  • Formulas

    • \text{Radian}= \frac{\pi}{180^{\circ}}\times\text{(degree)}

    • \text{Degree}=\frac{180^{\circ}}{\pi}\times\text{(radian)}

  • Approximations

    • 1\text{ rad}\approx57^{\circ}16'

    • 1^{\circ}\approx0.01746\text{ rad}

  • Common angles

    • 30^{\circ}=\tfrac{\pi}{6}; 45^{\circ}=\tfrac{\pi}{4}; 60^{\circ}=\tfrac{\pi}{3}; 90^{\circ}=\tfrac{\pi}{2}; 180^{\circ}=\pi; 270^{\circ}=\tfrac{3\pi}{2}; 360^{\circ}=2\pi

Worked Conversions

  • 40^{\circ}20' = \frac{121\pi}{540}\text{ rad}

  • 6\text{ rad} \approx 343^{\circ}38'11''

Arc–Angle–Radius Applications

  • Arc length formula l=r\theta led to example computations: finding radius for given arc & angle; tip travel of watch hand; ratio of radii when arcs equal etc.

Unit Circle Definitions (Radian Domain)

  • Any point P(a,b) on unit circle with arc x (radians) from A(1,0) satisfies

    • \cos x = a

    • \sin x = b

    • Identity \cos^2x+\sin^2x=1 (Pythagorean)

  • Periodicity: \sin(x+2n\pi)=\sin x; \cos(x+2n\pi)=\cos x for n\in\mathbb Z

  • Zeros

    • \sin x=0\;\Rightarrow\;x=n\pi

    • \cos x=0\;\Rightarrow\;x=(2n+1)\frac{\pi}{2}

Derived Functions

  • \tan x = \frac{\sin x}{\cos x}, undefined where \cos x=0

  • \cot x = \frac{\cos x}{\sin x}, undefined where \sin x=0

  • \sec x = \frac{1}{\cos x}, \csc x = \frac{1}{\sin x} with corresponding domain restrictions

  • Reciprocal/Pythagorean identities

    • 1+\tan^2x=\sec^2x

    • 1+\cot^2x=\csc^2x

Signs in Four Quadrants (ASTC Rule)

  • Quadrant I (0<x<\tfrac{\pi}{2}): all positive

  • Quadrant II (\tfrac{\pi}{2}<x<\pi): \sin,\csc positive

  • Quadrant III (\pi<x<\tfrac{3\pi}{2}): \tan,\cot positive

  • Quadrant IV (\tfrac{3\pi}{2}<x<2\pi): \cos,\sec positive

Domains & Ranges

  • y=\sin x,\cos x: domain \mathbb R; range [-1,1]

  • y=\tan x: domain \mathbb R\setminus{(2n+1)\tfrac{\pi}{2}}; range \mathbb R

  • y=\cot x: domain \mathbb R\setminus{n\pi}; range \mathbb R

  • y=\sec x: domain \mathbb R\setminus{(2n+1)\tfrac{\pi}{2}}; range (-\infty,-1]\cup[1,\infty)

  • y=\csc x: domain \mathbb R\setminus{n\pi}; range (-\infty,-1]\cup[1,\infty)

Monotonic Behaviour Within Quadrants (Snapshots)

  • In first quadrant \sin x rises 0\to1, \cos x falls 1\to0, \tan x rises 0\to\infty etc.

  • Second quadrant: \sin decreases to 0; \cos decreases 0\to-1; \tan increases -\infty\to0

  • Third quadrant: \sin negative decreasing 0\to-1; \tan rises 0\to\infty etc.

  • Fourth quadrant mirrors accordingly; notes on \pm\infty signify unbounded growth near vertical asymptotes

Special Values Table (radians given)

\begin{array}{c|ccccccc}
x & 0 & \tfrac{\pi}{6} & \tfrac{\pi}{4} & \tfrac{\pi}{3} & \tfrac{\pi}{2} & \tfrac{3\pi}{2} & 2\pi\\hline
\sin x & 0 & \tfrac{1}{2} & \tfrac{\sqrt3}{2} & 1 & 0 & -1 & 0\
\cos x & 1 & \tfrac{\sqrt3}{2} & \tfrac{1}{2} & 0 & -1 & 0 & 1\
\tan x & 0 & \tfrac{1}{\sqrt3} & 1 & \sqrt3 & \text{–} & 0 & 0\\end{array}
(‘–’ means undefined)

Fundamental Angle Transformations

  • Even/Odd: \cos(-x)=\cos x; \sin(-x)=-\sin x

  • Co–function: \sin\bigl(\tfrac{\pi}{2}-x\bigr)=\cos x; \cos\bigl(\tfrac{\pi}{2}-x\bigr)=\sin x

  • Supplementary etc.

    • \sin(\pi-x)=\sin x ; \cos(\pi-x)=-\cos x

    • \sin(\pi+x)=-\sin x ; \cos(\pi+x)=-\cos x

    • \sin(2\pi-x)=-\sin x ; \cos(2\pi-x)=\cos x

Sum & Difference Identities

  • \cos(x+y)=\cos x\cos y-\sin x\sin y

  • \cos(x-y)=\cos x\cos y+\sin x\sin y

  • \sin(x+y)=\sin x\cos y+\cos x\sin y

  • \sin(x-y)=\sin x\cos y-\cos x\sin y

  • Tangent (provided denominators non-zero)

    • \tan(x!+!y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}

    • \tan(x!-!y)=\dfrac{\tan x-\tan y}{1+\tan x\tan y}

  • Cotangent

    • \cot(x!+!y)=\dfrac{\cot x\cot y-1}{\cot y+\cot x}

    • \cot(x!-!y)=\dfrac{\cot x\cot y+1}{\cot y-\cot x}

Double, Triple & Half Angle

  • Double

    • \cos2x=\cos^{2}x-\sin^{2}x=2\cos^{2}x-1=1-2\sin^{2}x=\dfrac{1-\tan^{2}x}{1+\tan^{2}x}

    • \sin2x=2\sin x\cos x=\dfrac{2\tan x}{1+\tan^{2}x}

    • \tan2x=\dfrac{2\tan x}{1-\tan^{2}x}

  • Triple

    • \sin3x=3\sin x-4\sin^{3}x

    • \cos3x=4\cos^{3}x-3\cos x

    • \tan3x=\dfrac{3\tan x-\tan^{3}x}{1-3\tan^{2}x}

  • Half-angle (from double-angle inversion)

    • \sin\tfrac{x}{2}=\pm\sqrt{\dfrac{1-\cos x}{2}}

    • \cos\tfrac{x}{2}=\pm\sqrt{\dfrac{1+\cos x}{2}}

    • \tan\tfrac{x}{2}=\dfrac{\sin x}{1+\cos x}=\dfrac{1-\cos x}{\sin x}
      (Sign depends on quadrant of x/2)

Sum ↔ Product Transformations

  1. Product to Sum

    • 2\cos x\cos y=\cos(x+y)+\cos(x-y)

    • -2\sin x\sin y=\cos(x+y)-\cos(x-y)

    • 2\sin x\cos y=\sin(x+y)+\sin(x-y)

    • 2\cos x\sin y=\sin(x+y)-\sin(x-y)

  2. Sum to Product (by reversing above)

    • \cos x+\cos y=2\cos\tfrac{x+y}{2}\cos\tfrac{x-y}{2}

    • \cos x-\cos y=-2\sin\tfrac{x+y}{2}\sin\tfrac{x-y}{2}

    • \sin x+\sin y=2\sin\tfrac{x+y}{2}\cos\tfrac{x-y}{2}

    • \sin x-\sin y=2\cos\tfrac{x+y}{2}\sin\tfrac{x-y}{2}

Illustrative Worked Examples (selection)

  • Example 6 (given \cos x=-\tfrac35 in III quadrant) ⇒ \sin x=-\tfrac45, \tan x=\tfrac43, reciprocals accordingly

  • Example 8 \sin\bigl(\tfrac{31\pi}{3}\bigr)=\sin\bigl(10\pi+\tfrac{\pi}{3}\bigr)=\tfrac{\sqrt3}{2} (periodicity)

  • Example 11 \sin15^{\circ}=\sin(45^{\circ}-30^{\circ})=\tfrac{\sqrt6-\sqrt2}{4}

  • Example 20 Evaluation \tan\tfrac{\pi}{8}=\sqrt2-1 via double-angle quadratic

  • Example 21 Half-angle with negative cosine: if \tan x=\tfrac{3\sqrt3}{3} etc., derived \sin\tfrac{x}{2}=\tfrac{3}{\sqrt{10}},\;\cos\tfrac{x}{2}=-\tfrac1{\sqrt{10}}

Standard Exercise Sets (3.1–3.3)

  • 3.1: Convert between degrees & radians; arc–radius questions

  • 3.2: Determine remaining trig ratios given one and quadrant; evaluate circular-function arguments exceeding 2\pi or negatives

  • 3.3: Proof-oriented identities utilising sum/difference, double angles, etc.

Additional Key Identities from Chapter Summary

  • \cos4x=1-8\sin^{2}x\cos^{2}x

  • \cos6x=32\cos^{6}x-48\cos^{4}x+18\cos^{2}x-1

  • Cotangent cyclic product: \cot x\cot2x-\cot2x\cot3x-\cot3x\cot x=1

Graphical Insights (conceptual)

  • \sin x,\cos x periodic with period 2\pi (smooth wave)

  • \tan x,\cot x period \pi with vertical asymptotes where denominator zero

  • \sec x,\csc x share 2\pi period; resemble reciprocal envelopes of sine & cosine with asymptotes at their zeros

Historical Context

  • Indian mathematicians (Aryabhata 476–550, Brahmagupta 598, Bhaskara I 600 & II 1114) pioneered systematic trigonometry, sine tables, formulas beyond 90^{\circ}, early proofs of \sin(A+B).

  • Knowledge transmitted from India → Middle East → Europe; Greek methods replaced by concise Indian sine approach.

  • Herschel (1813) suggested inverse-function notation \sin^{-1}x.

  • Thales (600 BCE) used similar triangles to measure pyramids’ heights and sea distances; height-distance problems remain classical applications.

Practical & Ethical Connections

  • Precise angle measurement underpins GPS, satellite communication, architecture safety calculations.

  • Ethical diligence: engineering mis-calculations of trigonometric data can jeopardise lives (bridges, towers, medical imaging).

Quick Reference (Cheat-Sheet)

  • Arc length: l=r\theta (\theta in rad)

  • Periods: \sin,\cos,\sec,\csc\;\to2\pi; \tan,\cot\to\pi

  • Special numbers: \sqrt2\approx1.414, \sqrt3\approx1.732, \pi\approx3.1416

  • Exact radicals for 18°, 36°, 54°, 72° derivable via golden-ratio constructions (Bhāskara II) – often useful in polygon problems.

These bullet-point notes encapsulate every concept, example, formula, table, historical remark, and implication presented in the transcript, offering a comprehensive standalone study guide for Trigonometric Functions.