In-Depth Notes on Trig Identities

Understanding Trig Identities

  • Trig identities are expressions that remain true for all values of the angle $\theta$.

  • Example: The identity tan(θ)=cos(θ)sin(θ)​ is a basic example.

Pythagorean Identity

  • Derived from the unit circle where:

    • (a,b)=(cos(θ),sin(θ))

  • The basic relationship is given by the Pythagorean theorem:

    • cos2(θ)+sin2(θ)=1

  • This identity allows for rearrangement:

    • sin2(θ)=1−cos2(θ) or cos2(θ)=1−sin2(θ) are also valid.

Other Fundamental Trig Identities

  1. Basic Identities:

    • tan(θ)=cos(θ)sin(θ)​

  2. Negative Angle Identities:

    • sin(−θ)=−sin(θ)

    • cos(−θ)=cos(θ)

    • tan(−θ)=−tan(θ)

  3. Angle Sum and Difference Identities:

    • For sine:

      • sin(θ+ϕ)=sin(θ)cos(ϕ)+cos(θ)sin(ϕ)

      • sin(θ−ϕ)=sin(θ)cos(ϕ)−cos(θ)sin(ϕ)

    • For cosine:

      • cos(θ+ϕ)=cos(θ)cos(ϕ)−sin(θ)sin(ϕ)

      • cos(θ−ϕ)=cos(θ)cos(ϕ)+sin(θ)sin(ϕ)

Unit Circle and Trig Ratios

  • A unit circle helps to illustrate the values of sinθ and cosθ for various angles.

  • This geometric representation illustrates the basis for finding trig values and relationships.

Practical Applications of Trig Identities

  • Trig identities can simplify complex expressions or help solve equations involving angles, particularly in applications involving rotations, wave functions, and harmonic motion.

  • Example Application: Use of Pythagorean identity to verify that the expression remains true under various angles by substituting those angles into the equation.

  • Calculator usage example: Enter sin2(θ)+cos2(θ)=1 to demonstrate the identity.

Summary of Key Trig Identities

  • Important Trig Identities:

    • Pythagorean Identity:

    sin⁡2(θ)+cos⁡2(θ)=1

    • Definition of Tangent:

    tan⁡(θ)=sin⁡(θ)cos⁡(θ)

    • Sine of a Negative Angle:

    sin⁡(−θ)=−sin⁡(θ)

    • Cosine of a Negative Angle:

    cos⁡(−θ)=cos⁡(θ)

    • Angle Sum/Difference rules as discussed improve our understanding of combining angles' effects with respect to their trig values.

  • Note that these identities serve both in theoretical proofs and practical problem-solving.