Trigonometry Lecture Notes

Overview

• Focus on applying the concepts of double and compound angles.
• Practice solving trigonometric identities and simplifying expressions without calculators.

Compound Angle Formulas

• cos(a + b) Formula:
  • ( \cos(a + b) = \cos a \cos b - \sin a \sin b )

Example Problem

• Simplify ( \cos(180 - x) ):

  • Cosine Reduction: ( \cos(180 - x) = -\cos x ) (since cosine is negative in the second quadrant).
  • Sine Reduction: ( \sin(180 - x) = \sin x ).

• Expression ( \cos(180-x)\cos(180-x) - \sin(180-x)\sin(180-x) ) becomes:

  • ( (-\cos x)(-\cos x) - (\sin x)(\sin x) )
  • Simplifies to ( \cos^2 x - \sin^2 x )

• Recognize ( \cos^2 x - \sin^2 x ) as part of ( \cos 2x ) according to the formula:

  • ( \cos 2x = \cos^2 x - \sin^2 x )

Special Angles without a Calculator

• Cos 75° Example
  • ( 75° = 45° + 30° )
  • Use the compound angle formula:
    • ( \cos(45° + 30°) = \cos 45° \cos 30° - \sin 45° \sin 30° )

Special Triangles

• 30-60-90 Triangle:

  • Cosine and sine values derived from positions in triangle.

• 45-45-90 Triangle:

  • Isosceles triangle, ( \cos 45° = \sin 45° = \frac{1}{\sqrt{2}} )

Calculations

• ( \cos 45° = \frac{1}{\sqrt{2}} )

• ( \cos 30° = \frac{\sqrt{3}}{2} )

• ( \sin 45° = \frac{1}{\sqrt{2}} )

• ( \sin 30° = \frac{1}{2} )

• Calculated result:

  • ( \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} - 1}{2\sqrt{2}} )

Verification without Calculator

• Use a calculator to verify decimal equivalence.
• The manual calculation will match the decimal output of ( \cos 75° ) when using the calculator.

Key Takeaways

• For exams, know the special triangles and angle identities.
• Use compound angle formulas to simplify trigonometric expressions.
• Verifying results with a calculator involves checking decimal equivalence, but original steps should be performed manually.