Trigonometric Formulas and Equations

Double-Angle Formulas

  • Rewriting Expressions
    • Example: Rewrite extcos2(x)−12ext{cos}^2(x) - \frac{1}{2}
    • Apply the double-angle formula: extcos(2x)=2extcos2(x)−1ext{cos}(2x) = 2 ext{cos}^2(x) - 1extcos2(x)=1+extcos(2x)2ext{cos}^2(x) = \frac{1 + ext{cos}(2x)}{2}
      • Substitute into the expression when necessary.

Half-Angle Formulas

  • Formulas
    • extsin(x2)=extsin(x)extsqrt(1+extcos(x))ext{sin}\bigg(\frac{x}{2}\bigg) = \frac{ ext{sin}(x)}{ ext{sqrt}(1+ ext{cos}(x))}
    • extcos(x2)=extcos(x)extsqrt(1+extcos(x))ext{cos}\bigg(\frac{x}{2}\bigg) = \frac{ ext{cos}(x)}{ ext{sqrt}(1+ ext{cos}(x))}
    • exttan(x2)=1−extcos(x)extsin(x)ext{tan}\bigg(\frac{x}{2}\bigg) = \frac{1 - ext{cos}(x)}{ ext{sin}(x)}

Finding Exact Values

  • Example: Find the exact value of extcos(157.5exto)ext{cos}(157.5^ ext{o})
    • Rewrite using half-angle formulas:
    • extcos(157.5exto)=extcos(315exto/2)ext{cos}(157.5^ ext{o}) = ext{cos}(315^ ext{o}/2)
    • Apply half-angle formula:
      extcos(315exto)=1+extcos(315exto)2ext{cos}(315^ ext{o}) = \frac{1 + ext{cos}(315^ ext{o})}{2}

Trigonometric Equations

  • Basic Equations
    • Example: 2x+1=02x + 1 = 0
    • General Solution:
      x=−12x = -\frac{1}{2}
    • Example: 2extsin(x)−1=02 ext{sin}(x)-1=0
    • Solve for extsin(x)=12ext{sin}(x) = \frac{1}{2}
    • Solutions in quadrants I and II:
      x=extpi6,5extpi6x = \frac{ ext{pi}}{6}, \frac{5 ext{pi}}{6}

More Complex Equations

  • Example: 3extsin(x)+1=extsin(x)3 ext{sin}(x) + 1 = ext{sin}(x)
    • Rearrange to:
    • 2extsin(x)+1=02 ext{sin}(x) + 1 = 0
    • Find Quadrants for Negative Values
    • ( ext{sin}(x) = - rac{1}{2})
      ightarrow x = rac{7 ext{pi}}{6}, rac{11 ext{pi}}{6}

Law of Sines and Cosines

  • Law of Sines:

    • aextsin(A)=bextsin(B)=cextsin(C)\frac{a}{ ext{sin}(A)} = \frac{b}{ ext{sin}(B)} = \frac{c}{ ext{sin}(C)}
    • Used for non-right triangles with known angles/sides.

  • Example: Given A=40exto,B=60exto,a=12A=40^ ext{o}, B=60^ ext{o}, a=12:

    • Solve for the unknown side bb and angle CC.

Special Angle Identities

  • Use of Identities:
    • Formulas to compute exact values for angles that can compound into special angles.
    • Examples:
    • extsin(75exto)=extsin(30exto+45exto)ext{sin}(75^ ext{o}) = ext{sin}(30^ ext{o} + 45^ ext{o})

Reductions and Simplifications

  • Consolidating Functions:
    • Example: extcos(A)extcos(B)+extsin(A)extsin(B)ext{cos}(A) ext{cos}(B) + ext{sin}(A) ext{sin}(B) simplifies to extcos(A+B)ext{cos}(A+B).

Summary of Key Concepts

  • Ensure familiar with:
    • Double angle formulas, half-angle formulas, law of sines and cosines, and special angle identities.
  • Practice using these formulas to solve various trigonometric problems efficiently.