Lecture on Trigonometry for Calculus

Introduction

• Students often struggle with trigonometry in calculus due to the amount of memorization required.
• The goal is to understand basic concepts instead of memorizing everything.
• Sponsored by Brilliant.

The Unit Circle

• Concept: A circle with a radius of 1.
• Point Movement: Starts on the positive x-axis and rotates counterclockwise.
• Angle Theta: Denoted by ( \theta ) with horizontal and vertical components as ( \cos(\theta) ) and ( \sin(\theta) ) respectively.
• Right Triangle Definitions:
  • Adjacent = ( \cos(\theta) )
  • Opposite = ( \sin(\theta) )

Radians

• Definition: The full circle is ( 2\pi ) radians.
• Advantage: Matches the circumference of the unit circle, avoiding stretching factors in calculus.
• Calculator Tip: Always ensure it's set to radians.

Trigonometric Identities

• Pythagorean Identity: ( \cos^2(\theta) + \sin^2(\theta) = 1 )
• Graphs:
  • Cosine Graph: Reflects horizontal component.
  • Sine Graph: Reflects vertical component.
  • Graphs are similar, with sine lagging behind cosine.

Special Angles and Triangles

• ( \pi/4 ): Yields a 1-1-( \sqrt{2} ) triangle.
• ( \pi/6 ) and ( \pi/3 ): 1, ( \frac{1}{2} ), ( \frac{\sqrt{3}}{2} ) triangle.
• Rationalizing Denominators: Often unnecessary in higher mathematics.

Working with Quadrants

• Example: ( \frac{11\pi}{6} ) is in the fourth quadrant.
• Use known triangles to determine sine and cosine and adjust based on the quadrant.

Other Trigonometric Functions

• Definitions:
  • Tangent = ( \frac{\sin}{\cos} )
  • Cotangent, Secant, Cosecant are derived similarly.
• Graphs:
  • Tangent Graph: Derived from sine/cosine, noting where the cosine is zero (asymptotes).
  • Understanding asymptotes and zero crossings is key.

Additional Identities

• Other Pythagorean Identities:
  • ( 1 + \cot^2(\theta) = \csc^2(\theta) )
  • ( \tan^2(\theta) + 1 = \sec^2(\theta) )
• Function Pairings: Helpful in calculus

Geometric Meaning of Trig Functions

• Secant and Tangent: Can be visualized using extended triangles from the unit circle.
• Not often used but useful for understanding.

Key Trigonometric Identities

• Sum and Difference: Used for ( \sin(\alpha + \beta) ) and ( \cos(\alpha + \beta) ).
• Double Angle: Special cases of sum identities.
• Power Reduction: Useful for integrating powers.

Practical Application

• Understanding vs. Memorization: Knowing the concepts and being able to derive them is more useful than memorizing.
• Practice: Engage with problems for better retention.

Conclusion

• Further Learning: Practice and explore with resources like Brilliant.
• Encouragement: Engage with the material actively to improve understanding.