Trigonometric Identities: Double and Half Angle Formulas

Trigonometric Identities

  • Identities are formulas used for calculating different trigonometric values.
  • The lecture focuses on double angle and half angle identities.

Double Angle Formulas

  • These formulas are derived from the sum identities when both angles are the same (a = b).

Sine of Double Angle

  • Starting with the sum identity: \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
  • If a = b, then: \sin(2a) = \sin(a)\cos(a) + \sin(a)\cos(a) = 2\sin(a)\cos(a)

Cosine of Double Angle

  • Starting with the sum identity: \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)
  • If a = b, then: \cos(2a) = \cos^2(a) - \sin^2(a)
  • This form is similar to the Pythagorean identity.

Tangent of Double Angle

  • Starting with the sum identity: \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}
  • If a = b, then: \tan(2a) = \frac{2\tan(a)}{1 - \tan^2(a)}

Alternative Forms of the Cosine Double Angle Formula

  • Using the Pythagorean identity: \sin^2(a) + \cos^2(a) = 1

Version 1

  • Solve for \cos^2(a): \cos^2(a) = 1 - \sin^2(a)
  • Substitute into \cos(2a) = \cos^2(a) - \sin^2(a):
    \cos(2a) = (1 - \sin^2(a)) - \sin^2(a) = 1 - 2\sin^2(a)

Version 2

  • Solve for \sin^2(a): \sin^2(a) = 1 - \cos^2(a)
  • Substitute into \cos(2a) = \cos^2(a) - \sin^2(a):
    \cos(2a) = \cos^2(a) - (1 - \cos^2(a)) = 2\cos^2(a) - 1

Half Angle Formulas

  • Derived by manipulating the alternative forms of the cosine double angle formula.

Cosine of Half Angle

  • Starting with \cos(2a) = 2\cos^2(a) - 1
  • Solve for \cos(a):
    • Add 1: \cos(2a) + 1 = 2\cos^2(a)
    • Divide by 2: \frac{1 + \cos(2a)}{2} = \cos^2(a)
    • Take the square root: \cos(a) = \pm\sqrt{\frac{1 + \cos(2a)}{2}}
  • Replace a with \frac{\theta}{2}: \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}

Sine of Half Angle

  • Starting with \cos(2a) = 1 - 2\sin^2(a)
  • Solve for \sin(a):
    • Subtract 1: \cos(2a) - 1 = -2\sin^2(a)
    • Divide by -2: \frac{1 - \cos(2a)}{2} = \sin^2(a)
    • Take the square root: \sin(a) = \pm\sqrt{\frac{1 - \cos(2a)}{2}}
  • Replace a with \frac{\theta}{2}: \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}

Example: Finding \cos(22.5^{\circ})

  • Recognize that 22.5^{\circ} = \frac{45^{\circ}}{2}
  • Use the half-angle formula for cosine: \cos(22.5^{\circ}) = \cos(\frac{45^{\circ}}{2}) = \pm\sqrt{\frac{1 + \cos(45^{\circ})}{2}}
  • Since 22.5^{\circ} is in the first quadrant, cosine is positive.
  • \cos(45^{\circ}) = \frac{\sqrt{2}}{2}
  • Substitute: \cos(22.5^{\circ}) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}
  • Simplify:
    • \cos(22.5^{\circ}) = \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2}

Verification with Calculator

  • The example demonstrates how to use the half-angle formulas and verifies the result using a calculator.

Summary

  • The lecture covered double angle and half angle formulas for sine, cosine, and tangent.
  • These formulas are useful for finding trigonometric values of angles that are multiples or fractions of known angles.