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.