Lecture Notes on Trigonometric Functions and Identities

Key Concepts

• Trigonometric Functions: The lecture deals with the tangent, cosine, and sine functions, particularly focusing on their behavior in different quadrants and how they transform.
• Angle Reduction: Adjusting angles to fit within a common range (0 to 360 degrees) by adding or subtracting 360 degrees.
• Trigonometric Identities: Use of Pythagorean and co-function identities to simplify expressions.

Angle Adjustments

• When given an angle like tan(-12):

  • Convert it by adding 360 to get tan(348).
  • Recognize the quadrant of 348 degrees and apply the property that tan is negative in that quadrant, giving -tan(12).

• For cos(348):

  • Reduce it to cos(12).
  • Cosine is positive in this quadrant, so it remains cos(12).

• For angles less than 90 degrees (e.g., sin^2(10) and sin^2(80)), no reduction is needed.

Sine Squared Transformation

• Transform sin^2(321):
  • Convert to sin(39) considering it lies in a quadrant where sine is negative.
  • Squaring negates the negative, resulting in sin^2(39).

Expression Simplification

• Simplified expression:

  -tan(12) * cos(12) / (sin^2(10) + sin^2(80) - 2sin^2(39))

• Cosines cancel: cos(12) in the numerator and denominator cancels out.

• Recognize co-function identities:

  • sin(10) is equivalent to cos(80).
  • Use the identity sin^2(80) + cos^2(80) = 1 to simplify.

Further Simplification

• Results in 1 - 2sin^2(39).
• Recognize the identity for a double angle formula:
  • cos(2x) form where 2x = 78.

Final Steps

• Use co-function identity again:
  • cos(78) is equivalent to sin(12).
  • This leads to cancellation, simplifying to -1.

Conclusion

• The lecture demonstrates how to use angle reduction, trigonometric identities, and simplification techniques effectively to reduce complex trigonometric expressions to simpler forms.
• Emphasizes the importance of understanding the quadrant properties and identities to achieve simplification efficiently.