Sum and Difference Identities in Trigonometry

The Number One Mistake in Trigonometry

• The common mistake is assuming that the sine of a sum of angles is equal to the sum of their sines.
  sin(a + b) ≠ sin(a) + sin(b)
• This incorrect assumption would simplify trigonometry, but it's not valid.

Illustration with Sine of 75 Degrees

• To demonstrate the mistake, let's calculate sin(75°).
• 75 degrees can be expressed as 45° + 30°.
• If the initial assumption were correct:
  sin(75°) = sin(45°) + sin(30°)

Calculations Using Special Angles

• sin(45°) = sin(\frac{π}{4}) = \frac{\sqrt{2}}{2}
• sin(30°) = sin(\frac{π}{6}) = \frac{1}{2}
• Adding these:
  \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{\sqrt{2} + 1}{2}
• Approximation:
  \frac{1.4 + 1}{2} ≈ 1.2
• This would imply that sin(75°) = 1.2.

The Contradiction

• The sine of any angle cannot be greater than 1 because on the unit circle, the y-coordinate of any point is always less than or equal to 1.
• Therefore, the initial assumption sin(a + b) = sin(a) + sin(b) is incorrect.

The Correct Formula: Sine of a Sum Identity

• The correct identity for the sine of a sum is:
  sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Applying the Correct Formula to Sine of 75 Degrees

• Using the correct formula to find sin(75°)
  sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)

Calculations with the Sine of a Sum Identity

• sin(45°) = \frac{\sqrt{2}}{2}
• cos(30°) = \frac{\sqrt{3}}{2}
• cos(45°) = \frac{\sqrt{2}}{2}
• sin(30°) = \frac{1}{2}
• Therefore:
  sin(75°) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}

Sine of a Difference Identity

• The identity for the sine of a difference is:
  sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Example: Sine of 15 Degrees

• Let's calculate sin(15°), where 15° = 45° - 30°
• sin(15°) = sin(45°)cos(30°) - cos(45°)sin(30°) = \frac{\sqrt{6} - \sqrt{2}}{4}

Cosine of a Sum and Difference Identities

• Cosine of a sum:
  cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
• Cosine of a difference:
  cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Example: Cosine of 15 Degrees

• Using the cosine difference identity to find cos(15°), where 15° = 45° - 30°
• cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°) = \frac{\sqrt{6} + \sqrt{2}}{4}

Observation: cos(15°) and sin(75°) are Equal

• The result for cos(15°) is the same as sin(75°).
• Explanation: On the unit circle, 15° and 75° are mirror images about the line y = x, so their x and y coordinates (cosine and sine values) are switched.

Tangent of a Sum and Difference Identities

• Tangent of a sum:
  tan(a + b) = \frac{tan(a) + tan(b)}{1 - tan(a)tan(b)}
• Tangent of a difference:
  tan(a - b) = \frac{tan(a) - tan(b)}{1 + tan(a)tan(b)}

Example: Tangent of \frac{7π}{12}

• Let's find tan(\frac{7π}{12}), where \frac{7π}{12} = \frac{π}{4} + \frac{π}{3}
• tan(\frac{7π}{12}) = \frac{tan(\frac{π}{4}) + tan(\frac{π}{3})}{1 - tan(\frac{π}{4})tan(\frac{π}{3})}

Calculations and Result

• tan(\frac{π}{4}) = 1
• tan(\frac{π}{3}) = \sqrt{3}
• Therefore:
  tan(\frac{7π}{12}) = \frac{1 + \sqrt{3}}{1 - \sqrt{3}}

Conclusion

• Using sum and difference identities, sine, cosine and tangent can be calculated for a multitude of angles based on memorized angles on the unit circle.
• These identities work in both degrees and radians.