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.