Trigonometric Functions

• Introduction & Angles
  • Angle definition (Measure of rotation)
  • Systems of measurement
    • Degrees ($^\circ$)
    • Radians ($rad$)
  • Relation: $\pi$ radians = $180^\circ$
  • Arc length formula: $s = r\theta$
• Trigonometric Functions
  • Definition using Unit Circle
  • Six fundamental functions: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent
  • Sign of trigonometric functions in four quadrants (ASTC Rule)
  • Domain and Range of all functions
  • Periodicity and Graphs
    • Sine and Cosine graphs
    • Tangent graph
• Allied & Compound Angles
  • Trigonometric functions of allied angles ($-\theta$, $\pi \pm \theta$, $\frac{\pi}{2} \pm \theta$)
  • Compound angle formulas
    • $\sin(A \pm B)$
    • $\cos(A \pm B)$
    • $\tan(A \pm B)$
  • Multiple and Sub-multiple angles ($2\theta$, $\frac{\theta}{2}$)
    • $\sin 2\theta$, $\cos 2\theta$, $\tan 2\theta$
  • Transformation formulas
    • Sum to Product ($\sin C + \sin D$)
    • Product to Sum ($2\sin A \cos B$)
• Trigonometric Equations
  • Definition (Equations involving trig functions)
  • Principal Solutions ($\theta \in [0, 2\pi]$)
  • General Solutions
    • $\sin \theta = 0 \implies \theta = n\pi$
    • $\cos \theta = 0 \implies \theta = (2n + 1)\frac{\pi}{2}$
    • $\sin \theta = \sin \alpha \implies \theta = n\pi + (-1)^n\alpha$
    • $\cos \theta = \cos \alpha \implies \theta = 2n\pi \pm \alpha$
    • $\tan \theta = \tan \alpha \implies \theta = n\pi + \alpha$
• Sine and Cosine Formulae (Properties of Triangles)
  • Sine Formula: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
  • Cosine Formula
    • $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
  • Projection Formulae