Trigonometry Lecture Notes

Introduction to Trigonometry

• Trigonometry deals with right angle triangles.
• Studies relationships between triangle sides and angles.
• Can find unknown side lengths or angles using known values.

Triangle Basics

• Right Angle Triangle: Contains one right angle.
• Labeling Sides:
  • Hypotenuse (h): Longest side, opposite the right angle.
  • Opposite (o): Side opposite the angle (Theta).
  • Adjacent (a): Side next to the angle (Theta).

Trigonometric Functions

• Ratios based on the angle (Theta):
  • Sine (sin): ( ext{sin} heta = \frac{opposite}{hypotenuse} )
  • Cosine (cos): ( ext{cos} heta = \frac{adjacent}{hypotenuse} )
  • Tangent (tan): ( ext{tan} heta = \frac{opposite}{adjacent} )
• Mnemonic to remember:
  • Some Old Hags Can't Always Hack Their Old Age
    • sin = opposite/hypotenuse
    • cos = adjacent/hypotenuse
    • tan = opposite/adjacent

Example Problem 1

• Triangle with an angle of 35° and hypotenuse of 12 m.
• Find opposite side (x):
  • Use sine function:
  • ( ext{sin}(35°) = \frac{x}{12} )
  • Calculate ( ext{sin}(35°) \approx 0.57 ):
  • ( 0.57 = \frac{x}{12} )
  • Multiply: ( x = 12 * 0.57 = 6.88 m )

Example Problem 2

• Triangle with an angle of 48° and adjacent side of 15 m.
• Find opposite side (x):
  • Use tangent function:
  • ( ext{tan}(48°) = \frac{15}{x} )
  • Calculate ( ext{tan}(48°) \approx 1.11 ):
  • ( 1.11 = \frac{15}{x} )
  • Solve for x: ( x = \frac{15}{1.11} \approx 1.51 m )

Finding Angles

• Example Problem 3:
  • Given sides: 105 m (hypotenuse) and 33 m (opposite).
  • Find angle (Theta):
  • Use sine function:
  • ( ext{sin} \theta = \frac{33}{105} \approx 0.314 )
  • Use inverse sine to find angle: ( \theta = \text{sin}^{-1}(0.314) \approx 18.3° )

Example Problem 4

• Given sides: 17 m (hypotenuse) and 12 m (adjacent).
• Find angle (Theta):
  • Use cosine function:
  • ( ext{cos} \theta = \frac{12}{17} \approx 0.71 )
  • Use inverse cosine to find angle: ( \theta = \text{cos}^{-1}(0.71) \approx 45.1° )

Conclusion

• Trigonometry is straightforward with practice and familiarity with functions.
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