Trigonometric Functions — Chapter 3 (Comprehensive Notes & Short Answers) Summary & Study Notes

🔹 Introduction

Trigonometry (from Greek "trigon" + "metron") is the study of relationships between angles and lengths in triangles. Historically used for navigation, surveying and astronomy, it now appears in many fields such as seismology, signal processing and engineering.

📏 Angle Measurement: Degrees and Radians

• Degree: one revolution = $360^\circ$. One degree $= 60'$ (minutes), one minute $= 60''$ (seconds).
• Radian: an angle subtended at the centre of a circle by an arc of length equal to the radius. In a circle of radius $r$, arc length $l$ and central angle $\theta$ (in radians) satisfy $l = r\theta$.
• Relation between units: $2\pi$ radians $= 360^\circ$, so $\pi$ radians $= 180^\circ$. Thus to convert,
  • Radian $= \dfrac{\pi}{180^\circ} \times$ Degree,
  • Degree $= \dfrac{180^\circ}{\pi} \times$ Radian.

📐 Unit Circle and Definitions

• On the unit circle (radius 1) with a point $P(a,b)$ at angle $x$ (radians) from the positive $x$-axis, define cosine and sine by $\cos x = a$, $\sin x = b$.
• Pythagorean identity: $\cos^2 x + \sin^2 x = 1$ for all real $x$.
• Other functions: $\tan x = \dfrac{\sin x}{\cos x}$ (when $\cos x \neq 0$), $\cot x = \dfrac{\cos x}{\sin x}$, $\sec x = \dfrac{1}{\cos x}$, $\csc x = \dfrac{1}{\sin x}$ with their domain restrictions.

✅ Signs and Quadrants

• The unit-circle coordinates determine signs of trig functions in quadrants:
  • I (0 to $\pi/2$): all positive.
  • II ($\pi/2$ to $\pi$): sin/csc positive; cos/sec/tan negative/depends.
  • III ($\pi$ to $3\pi/2$): tan/cot positive; sin/cos negative.
  • IV ($3\pi/2$ to $2\pi$): cos/sec positive; sin/csc negative.
• Even/odd properties: $\cos(-x) = \cos x$, $\sin(-x) = -\sin x$.

🔁 Periodicity and Domain/Range

• Periodicity: $\sin$ and $\cos$ have period $2\pi$; $\tan$ and $\cot$ have period $\pi$.
• Ranges:
  • $\sin x, \cos x$: range $[-1,1]$.
  • $\tan x, \cot x$: range $(-\infty,\infty)$ (with excluded points in domain).
  • $\sec x, \csc x$: range $(-\infty,-1] \cup [1,\infty)$.

✳️ Key Identities (Sum/Difference and Consequences)

• Sum/difference formulas:
  • $\cos(x+y) = \cos x\cos y - \sin x\sin y$,
  • $\sin(x+y) = \sin x\cos y + \cos x\sin y$,
  • $\tan(x+y) = \dfrac{\tan x + \tan y}{1 - \tan x\tan y}$ (when denom $\neq 0$).
• Double-angle and related forms:
  • $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \dfrac{1 - \tan^2 x}{1 + \tan^2 x}$,
  • $\sin 2x = 2\sin x\cos x$,
  • $\tan 2x = \dfrac{2\tan x}{1 - \tan^2 x}$.
• Triple-angle examples:
  • $\sin 3x = 3\sin x - 4\sin^3 x$,
  • $\cos 3x = 4\cos^3 x - 3\cos x$.

🔄 Product-to-Sum and Sum-to-Product

• Useful transformations:
  • $\cos x + \cos y = 2\cos\dfrac{x+y}{2}\cos\dfrac{x-y}{2}$,
  • $\cos x - \cos y = -2\sin\dfrac{x+y}{2}\sin\dfrac{x-y}{2}$,
  • $\sin x + \sin y = 2\sin\dfrac{x+y}{2}\cos\dfrac{x-y}{2}$,
  • $\sin x - \sin y = 2\cos\dfrac{x+y}{2}\sin\dfrac{x-y}{2}$.

🧭 Arc Length and Applications

• Arc length formula: in a circle of radius $r$, an arc subtending angle $\theta$ (radians) has length $l = r\theta$.
• Example application: the minute hand of length $r$ moves through angle $\theta = \dfrac{\text{minutes}}{60} \times 2\pi$, so distance $= r\theta$.

📚 Summary (Essential Points)

• Convert degrees and radians via $\pi$ radians $= 180^\circ$.
• Define trig functions on the unit circle; remember domain restrictions for reciprocals.
• Memorise sum/difference and double/triple-angle formulas; use them to simplify expressions or evaluate exact values (e.g., $\sin 15^\circ$).