Trigonometry Basics Lecture Notes

Introduction to Trigonometry

• Focus: Understanding basic trigonometric ratios using the acronym SOHCAHTOA for right triangles.

SOHCAHTOA Explained

• S (Sine)
  • Sine ( \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} )
• C (Cosine)
  • Cosine ( \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
• T (Tangent)
  • Tangent ( \theta = \frac{\text{Opposite}}{\text{Adjacent}} )

Example 1: Right Triangle Ratios

• Triangle with sides 3, 4, 5
• Sine ( \theta = \frac{4}{5} )
• Cosine ( \theta = \frac{3}{5} )
• Tangent ( \theta = \frac{4}{3} )
• Tangent can also be calculated as ( \frac{\text{Sine}}{\text{Cosine}} )

Example 2: Right Triangle with Missing Side

• Given sides 5, 12; find hypotenuse using Pythagorean theorem:
  • ( c^2 = a^2 + b^2 )
  • ( c = 13 )
• Sine ( \theta = \frac{5}{13} )
• Cosine ( \theta = \frac{12}{13} )
• Tangent ( \theta = \frac{5}{12} )

Special Right Triangles

• Memorize common triangles:
  • 3-4-5
  • 5-12-13
  • 8-15-17
  • 7-24-25
  • 9-40-41 (less common)

Example 3: Special Right Triangle

• Given sides 14, 50; find missing side using 7-24-25 triangle (doubled):
  • Missing side is 48
• Sine ( \theta = \frac{14}{50} = \frac{7}{25} )
• Cosine ( \theta = \frac{48}{50} = \frac{24}{25} )
• Tangent ( \theta = \frac{14}{48} = \frac{7}{24} )

Reciprocal Trig Functions

• Cosecant (csc) = ( \frac{1}{\text{Sine}} )
• Secant (sec) = ( \frac{1}{\text{Cosine}} )
• Cotangent (cot) = ( \frac{1}{\text{Tangent}} )

Reference Angles and Quadrants

• Determine in which quadrant an angle lies:
  • Quadrant 1: All positive
  • Quadrant 2: Sine positive
  • Quadrant 3: Tangent positive
  • Quadrant 4: Cosine positive
• Reference Angle: Acute angle formed with the x-axis

Example Problems

• Find Cosine and Tangent given Sine:
  • Example: ( \sin \theta = \frac{8}{17} )
  • Use reference angles and the Pythagorean theorem to find missing sides

Converting Between Degrees and Radians

• ( \pi ) radians = 180 degrees
• Use conversion factor ( \frac{180}{\pi} ) or ( \frac{\pi}{180} )

Evaluating Trigonometric Functions without a Calculator

• Use special triangles or the unit circle
• 30-60-90 Triangle
  • Sine 30 = ( \frac{1}{2} )
  • Cosine 30 = ( \frac{\sqrt{3}}{2} )
• 45-45-90 Triangle
  • Cosine 45 = ( \frac{\sqrt{2}}{2} )

Summary

• Trigonometry relies heavily on understanding triangle properties and the relationship between angles and sides.
• Familiarity with special triangles and trigonometric identities is essential.

Additional Resources

• Check out the recommended trigonometry playlist for more examples and detailed explanations.