Lecture on Sum and Difference Formulas in Trigonometry

Sum and Difference Formulas

• Sine Sum Formula:
  • $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
• Sine Difference Formula:
  • $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$

Application: Evaluating $\sin(75^\circ)$

• Identify angles that sum to 75°: $30° + 45°$.
• Use the sine sum formula:
  • $a = 30°, b = 45°$
  • $\sin(75°) = \sin(30°)\cos(45°) + \cos(30°)\sin(45°)$
  • $\sin(30°) = \frac{1}{2}$, $\cos(45°) = \frac{\sqrt{2}}{2}$
  • $\cos(30°) = \frac{\sqrt{3}}{2}$, $\sin(45°) = \frac{\sqrt{2}}{2}$
  • Computation:
    • $\frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}$
    • $\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}$
    • Combined: $\frac{\sqrt{2} + \sqrt{6}}{4}$

Application: Evaluating $\sin(15^\circ)$

• Identify angles for difference: $45° - 30°$.
• Use the sine difference formula:
  • $a = 45°, b = 30°$
  • $\sin(15°) = \sin(45°)\cos(30°) - \cos(45°)\sin(30°)$
  • Computation:
    • $\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2}$
    • $\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$
    • Combined: $\frac{\sqrt{6} - \sqrt{2}}{4}$

Special Triangles

• 30-60-90 Triangle:
  • Sides: $1, \sqrt{3}, 2$
  • $\sin(30°) = \frac{1}{2}$, $\sin(60°) = \frac{\sqrt{3}}{2}$
  • $\cos(30°) = \frac{\sqrt{3}}{2}$
  • $\tan(30°) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
• 45-45-90 Triangle:
  • Sides: $1, 1, \sqrt{2}$
  • $\sin(45°) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
  • $\cos(45°) = \frac{\sqrt{2}}{2}$
  • $\tan(45°) = 1$

Converting Radians to Degrees

• Example: $\cos(\frac{7\pi}{12})$
  • Convert: $\frac{7\pi}{12} \times \frac{180°}{\pi}$
  • Result: $105°$
• Use angles that sum to $105°$: $60° + 45°$
• Use the cosine sum formula:
  • $\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$
  • Calculation: $\cos(60°)\cos(45°) - \sin(60°)\sin(45°)$
  • Results in: $\frac{\sqrt{2} - \sqrt{6}}{4}$

Tangent Sum and Difference Formulas

• Tangent Difference Formula:
  • $\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$
• Example: $\tan(\frac{\pi}{12})$
  • Convert: $\frac{15°}{\pi}$
  • Use angles for difference: $45° - 30°$
  • Calculation: $\tan(45°) - \tan(30°)$
  • Results in: $\frac{1 - \frac{\sqrt{3}}{3}}{1 + 1 \times \frac{\sqrt{3}}{3}}$
  • Simplified to: $2 - \sqrt{3}$

Complex Example: $\tan(\frac{23\pi}{12})$

• Convert: $\tan(345°)$
• Use angles for sum: $120° + 225°$
• Use tangent sum formula:
  • $\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}$
  • Calculation considering quadrants and reference angles
  • Simplified to: $\sqrt{3} - 2$

SOHCAHTOA Mnemonic

• SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$