Trig

Chapter 1  Trigonometric Functions

1-1 Angles 
1-2 Angle Relationships & Similar Triangles 
1-3 Trig Functions 
1-4 Using Definitions

  • Angle basics: positive = counterclockwise, negative = clockwise. Standard position: vertex at origin, initial side on +x-axis. Coterminal: \theta \pm 360\degree (or \theta \pm 2\pi).
  • Quadrantal angles: multiples of 90\degree (\pi/2).
  • Similar triangles: same angles \Rightarrow side ratios equal. This justifies trig ratios.
  • Right-triangle definitions (acute \theta):
    • \sin \theta=\frac{\text{opp}}{\text{hyp}}, \cos \theta=\frac{\text{adj}}{\text{hyp}}, \tan \theta=\frac{\text{opp}}{\text{adj}}.
    • Reciprocals: \csc=1/\sin, \sec=1/\cos, \cot=1/\tan .
  • Pythagorean identity: \sin^2\theta+\cos^2\theta=1. Also 1+\tan^2\theta=\sec^2\theta, 1+\cot^2\theta=\csc^2\theta .
  • Signs by quadrant (ASTC): QI: all +; QII: sin +; QIII: tan +; QIV: cos +.
  • Cofunction idea (acute): \sin\theta=\cos(90^\circ-\theta) , etc.

Chapter 2  Acute Angles & Right Triangles

2-1 Trig of Acute Angles 
2-2 Non-Acute Angles 
2-3 Approximations 
2-4–2-5 Right-Triangle Apps

  • Special angles: 30^\circ,45^\circ,60^\circ values (memorize exact \sin,\cos,\tan).
  • Extending to any angle: use reference angles and quadrant signs to evaluate \sin,\cos,\tan for non-acute angles.
  • Approximations: calculator in degree mode here; watch rounding and mode.
  • Solve right triangles: need one side + one acute angle (or two sides).
    • Common formulas: \text{opp} = (\tan \theta)\,\text{adj}, \text{adj} = (\cot \theta)\,\text{opp}, \text{hyp} = \frac{\text{opp}}{\sin\theta} = \frac{\text{adj}}{\cos\theta} .
  • Applications: height/distances using angle of elevation/depression; bearings; navigation.
  • Problem flow: draw, mark knowns, pick a ratio, solve symbolically, plug numbers, label units.

Chapter 3  Radian Measure & the Unit Circle

3-1 Radian Measure 
3-2 Applications 
3-3 Unit Circle & Circular Functions 
3-4 Linear/Angular Speed

  • Degrees \leftrightarrow radians: 180^\circ=\pi\ \text{rad}. Convert: x^\circ!\cdot\frac{\pi}{180}=\text{rad}; x\,\text{rad}!\cdot\frac{180}{\pi}=\text{deg}.
  • Arc length: s=r\theta (\theta in radians).
  • Sector area: A=\tfrac12 r^2\theta .
  • Unit circle values: exact (\cos\theta,\sin\theta) at multiples of 30^\circ( \pi/6), 45^\circ(\pi/4), 60^\circ(\pi/3) and quadrantal angles.
  • Circular (unit-circle) definitions:
    \sin\theta=y, \cos\theta=x, \tan\theta=y/x on the unit circle.
  • Angular & linear speed: \omega=\frac{\theta}{t} (rad/s); v=r\omega. Use for gears, pulleys, turntables.
  • Periodicity: \sin,\cos period 2\pi; \tan,\cot period \pi.

Chapter 4  Graphs of the Circular Functions

4-1 Sine/Cosine Graphs 
4-2 Translations 
4-3 Tangent/Cotangent 
4-4 Secant/Cosecant 
4-5 Harmonic Motion

  • Base graphs: \sin x and \cos x oscillate between \left[-1,1\right] with period 2\pi.
  • Transformations: y=A\sin(B(x-C))+D (same for cos).
    • Amplitude =|A| (vertical stretch).
    • Period =\frac{2\pi}{|B|} (for sin/cos).
    • Phase shift =C (right if C>0).
    • Vertical shift =D.
  • Tangent/Cotangent: vertical asymptotes where \cos x=0 (for tan) or \sin x=0 (for cot); period =\frac{\pi}{|B|} .
  • Secant/Cosecant: reciprocals of cos/sin; use the corresponding sine/cosine “guides” to place U-shaped branches and asymptotes.
  • Harmonic motion: x(t)=A\cos(\omega t+\phi) or A\sin(\omega t+\phi).
    • Amplitude =|A|; angular frequency \omega; frequency f=\frac{\omega}{2\pi}; period T=\frac{2\pi}{\omega} .
    • Simple models: springs, pendulums (small angles), AC voltage.

Quick checklists & common pitfalls

  • Mode checks: Degrees for Ch. 2 word problems; Radians for Ch. 3 formulas s=r\theta, A=\tfrac12 r^2\theta .
  • Reference angles: Always pair with quadrant signs (ASTC).
  • Graphs: Identify A,B,C,D before plotting; mark one full period.
  • Units: Keep r and s in the same units; \theta must be radians in arc/sector formulas.
  • Exact values first: Know the 30-45-60 triangle and unit-circle points cold.