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.s i n ( a + b ) ≠ s i n ( a ) + s i n ( b ) sin(a + b) ≠ sin(a) + sin(b) s in ( a + b ) = s in ( a ) + s in ( 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 s i n ( 75 ° ) sin(75°) s in ( 75° ) 75 degrees can be expressed as 45 ° + 30 ° 45° + 30° 45° + 30° If the initial assumption were correct:s i n ( 75 ° ) = s i n ( 45 ° ) + s i n ( 30 ° ) sin(75°) = sin(45°) + sin(30°) s in ( 75° ) = s in ( 45° ) + s in ( 30° ) Calculations Using Special Angles s i n ( 45 ° ) = s i n ( π 4 ) = 2 2 sin(45°) = sin(\frac{π}{4}) = \frac{\sqrt{2}}{2} s in ( 45° ) = s in ( 4 π ) = 2 2 s i n ( 30 ° ) = s i n ( π 6 ) = 1 2 sin(30°) = sin(\frac{π}{6}) = \frac{1}{2} s in ( 30° ) = s in ( 6 π ) = 2 1 Adding these:2 2 + 1 2 = 2 + 1 2 \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{\sqrt{2} + 1}{2} 2 2 + 2 1 = 2 2 + 1 Approximation:1.4 + 1 2 ≈ 1.2 \frac{1.4 + 1}{2} ≈ 1.2 2 1.4 + 1 ≈ 1.2 This would imply that s i n ( 75 ° ) = 1.2 sin(75°) = 1.2 s in ( 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 s i n ( a + b ) = s i n ( a ) + s i n ( b ) sin(a + b) = sin(a) + sin(b) s in ( a + b ) = s in ( a ) + s in ( b ) The correct identity for the sine of a sum is:s i n ( a + b ) = s i n ( a ) c o s ( b ) + c o s ( a ) s i n ( b ) sin(a + b) = sin(a)cos(b) + cos(a)sin(b) s in ( a + b ) = s in ( a ) cos ( b ) + cos ( a ) s in ( b ) Using the correct formula to find s i n ( 75 ° ) sin(75°) s in ( 75° ) s i n ( 75 ° ) = s i n ( 45 ° ) c o s ( 30 ° ) + c o s ( 45 ° ) s i n ( 30 ° ) sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°) s in ( 75° ) = s in ( 45° ) cos ( 30° ) + cos ( 45° ) s in ( 30° ) Calculations with the Sine of a Sum Identity s i n ( 45 ° ) = 2 2 sin(45°) = \frac{\sqrt{2}}{2} s in ( 45° ) = 2 2 c o s ( 30 ° ) = 3 2 cos(30°) = \frac{\sqrt{3}}{2} cos ( 30° ) = 2 3 c o s ( 45 ° ) = 2 2 cos(45°) = \frac{\sqrt{2}}{2} cos ( 45° ) = 2 2 s i n ( 30 ° ) = 1 2 sin(30°) = \frac{1}{2} s in ( 30° ) = 2 1 Therefore:s i n ( 75 ° ) = 2 2 ⋅ 3 2 + 2 2 ⋅ 1 2 = 6 + 2 4 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} s in ( 75° ) = 2 2 ⋅ 2 3 + 2 2 ⋅ 2 1 = 4 6 + 2 Sine of a Difference Identity The identity for the sine of a difference is:s i n ( a − b ) = s i n ( a ) c o s ( b ) − c o s ( a ) s i n ( b ) sin(a - b) = sin(a)cos(b) - cos(a)sin(b) s in ( a − b ) = s in ( a ) cos ( b ) − cos ( a ) s in ( b ) Example: Sine of 15 Degrees Let's calculate s i n ( 15 ° ) sin(15°) s in ( 15° ) 15 ° = 45 ° − 30 ° 15° = 45° - 30° 15° = 45° − 30° s i n ( 15 ° ) = s i n ( 45 ° ) c o s ( 30 ° ) − c o s ( 45 ° ) s i n ( 30 ° ) = 6 − 2 4 sin(15°) = sin(45°)cos(30°) - cos(45°)sin(30°) = \frac{\sqrt{6} - \sqrt{2}}{4} s in ( 15° ) = s in ( 45° ) cos ( 30° ) − cos ( 45° ) s in ( 30° ) = 4 6 − 2 Cosine of a Sum and Difference Identities Cosine of a sum:c o s ( a + b ) = c o s ( a ) c o s ( b ) − s i n ( a ) s i n ( b ) cos(a + b) = cos(a)cos(b) - sin(a)sin(b) cos ( a + b ) = cos ( a ) cos ( b ) − s in ( a ) s in ( b ) Cosine of a difference:c o s ( a − b ) = c o s ( a ) c o s ( b ) + s i n ( a ) s i n ( b ) cos(a - b) = cos(a)cos(b) + sin(a)sin(b) cos ( a − b ) = cos ( a ) cos ( b ) + s in ( a ) s in ( b ) Example: Cosine of 15 Degrees Using the cosine difference identity to find c o s ( 15 ° ) cos(15°) cos ( 15° ) 15 ° = 45 ° − 30 ° 15° = 45° - 30° 15° = 45° − 30° c o s ( 15 ° ) = c o s ( 45 ° ) c o s ( 30 ° ) + s i n ( 45 ° ) s i n ( 30 ° ) = 6 + 2 4 cos(15°) = cos(45°)cos(30°) + sin(45°)sin(30°) = \frac{\sqrt{6} + \sqrt{2}}{4} cos ( 15° ) = cos ( 45° ) cos ( 30° ) + s in ( 45° ) s in ( 30° ) = 4 6 + 2 Observation: c o s ( 15 ° ) cos(15°) cos ( 15° ) s i n ( 75 ° ) sin(75°) s in ( 75° ) The result for c o s ( 15 ° ) cos(15°) cos ( 15° ) s i n ( 75 ° ) sin(75°) s in ( 75° ) Explanation: On the unit circle, 15° and 75° are mirror images about the line y = x y = x y = x Tangent of a Sum and Difference Identities Tangent of a sum:t a n ( a + b ) = t a n ( a ) + t a n ( b ) 1 − t a n ( a ) t a n ( b ) tan(a + b) = \frac{tan(a) + tan(b)}{1 - tan(a)tan(b)} t an ( a + b ) = 1 − t an ( a ) t an ( b ) t an ( a ) + t an ( b ) Tangent of a difference:t a n ( a − b ) = t a n ( a ) − t a n ( b ) 1 + t a n ( a ) t a n ( b ) tan(a - b) = \frac{tan(a) - tan(b)}{1 + tan(a)tan(b)} t an ( a − b ) = 1 + t an ( a ) t an ( b ) t an ( a ) − t an ( b ) Example: Tangent of 7 π 12 \frac{7π}{12} 12 7 π Let's find t a n ( 7 π 12 ) tan(\frac{7π}{12}) t an ( 12 7 π ) 7 π 12 = π 4 + π 3 \frac{7π}{12} = \frac{π}{4} + \frac{π}{3} 12 7 π = 4 π + 3 π t a n ( 7 π 12 ) = t a n ( π 4 ) + t a n ( π 3 ) 1 − t a n ( π 4 ) t a n ( π 3 ) tan(\frac{7π}{12}) = \frac{tan(\frac{π}{4}) + tan(\frac{π}{3})}{1 - tan(\frac{π}{4})tan(\frac{π}{3})} t an ( 12 7 π ) = 1 − t an ( 4 π ) t an ( 3 π ) t an ( 4 π ) + t an ( 3 π ) Calculations and Result t a n ( π 4 ) = 1 tan(\frac{π}{4}) = 1 t an ( 4 π ) = 1 t a n ( π 3 ) = 3 tan(\frac{π}{3}) = \sqrt{3} t an ( 3 π ) = 3 Therefore:t a n ( 7 π 12 ) = 1 + 3 1 − 3 tan(\frac{7π}{12}) = \frac{1 + \sqrt{3}}{1 - \sqrt{3}} t an ( 12 7 π ) = 1 − 3 1 + 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.