Trigonometric Identities and Their Applications

Introduction

• Focus on trigonometric identities and solving trigonometric equations.
• Importance of memorizing basic trigonometric identities despite having notes available.
• Key identities: reciprocal, quotient, Pythagorean, double angle, sum and difference, cofunctions, half angle.
• Emphasis on reduction formulas, particularly useful in calculus (Calc 2 and Calc 3).
• Sum-to-product and product-to-sum formulas exist but are less emphasized.

Key Trigonometric Identities

1. Reciprocal Identities
2. Quotient Identities
3. Pythagorean Identities
4. Double Angle Identities
5. Sum and Difference Identities
6. Cofunction Identities
7. Half Angle Identities
8. Reduction Formulas
   • Particularly crucial for calculus problems.

Example Problems

Finding Exact Values Using Identities

1. Cosine of π/12

   • Use sum and difference identity: ( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta )
   • Break down into known angles: ( \pi/4 - \pi/6 )
   • Calculated exact value: ( \frac{\sqrt{6} + \sqrt{2}}{4} )

2. Cosine of 15 Degrees Using Half Angle Formula

   • Half angle identity: ( \cos(\alpha/2) = \pm \sqrt{(1 + \cos \alpha)/2} )
   • Break down into known angle: ( 30^\circ )
   • Calculated exact value: ( \frac{\sqrt{2 + \sqrt{3}}}{2} )

Double Angle Identities

1. Given ( \sin \phi = 5/8 ) and ( \phi ) in ( [0, \pi/2] )
   • Calculate ( \sin 2\phi, \cos 2\phi, \text{and} \tan 2\phi )
   • Use double angle formulas:
     • ( \sin(2\phi) = 2 \sin \phi \cos \phi )
     • ( \cos(2\phi) = \cos^2 \phi - \sin^2 \phi )
     • ( \tan(2\phi) = \frac{2 \tan \phi}{1 - \tan^2 \phi} )
   • Calculate values using a constructed triangle and Pythagorean theorem.

Reduction Formula

1. Reduce ( \cos^4 x ) using reduction formula
   • Break into ( \cos^2 x \times \cos^2 x )
   • Use reduction formula: ( \cos^2 x = \frac{1 + \cos 2x}{2} )
   • Calculate reduced form: ( \frac{3}{2} + 2 \cos 2x + \frac{1}{2} \cos 4x )

Conclusion

• Understanding trigonometric identities is crucial for solving equations and reducing expressions, especially in advanced calculus.
• Future topics will explore additional identities and their applications.