Addition and Subtraction Formulas in Trigonometry

Addition and Subtraction Formulas in Trigonometry:

unit circle to explain Addition and Subtraction Formulas in Trigonometry

Let us consider a unit circle with its centre at O, the origin of the coordinate system. Then, the coordinates of P, the point of intersection of the circle with the positive X-axis are (1, 0). If ∠POQ = A, ∠QOR = B and ∠POS = -B, then the coordinates of Q, R and S are (cos A, sin A), [cos (A + B), sin (A + B)] and (cos B, -sin B) respectively. Since ∠POR = A + B and ∠QOS = A + B, we have PR = QS.

⇒ PR2 = QS2
⇒ {cos (A + B) – 1}2 + {sin (A + B) – 0}2 = (cos A – cos B)2 + (sin A + sin B)2
⇒ cos2 (A + B) + 1 – 2 cos (A + B) + sin2 (A + B) = cos2 A + cos2 B – 2cos A cos B + sin2 A + sin2 B + 2 sin A sin B
⇒ 1 + 1 – 2 cos (A + B) = 2 – 2 cos A cos B + 2 sin A sin B
⇒ -2 cos (A + B) = -2 (cos A cos B – sin A sin B)
⇒ cos (A + B) = cos A cos B – sin A sin B …………(i)

Replacing B by -B, we get-
cos (A – B) = cos A cos (-B) – sin A sin (-B) = cos A cos B + sin A sin B [∵ cos (-B) = cos B and sin (-B) = -sin B] ……………(ii)

Now, replacing A by (π/2 + A) in (i), we get-
cos (π/2 + A + B) = cos (π/2 + A) cos B – sin (π/2 + A) sin B
⇒ -sin (A + B) = -sin A cos B – cos A sin B
⇒ sin (A + B) = sin A cos B + cos A sin B …………(iii)

Replacing B by -B in (iii), we get-
sin (A – B ) = sin A cos (-B) + cos A sin (-B) = sin A cos B – cos A sin B ………(iv)
Trigonometry Addition and Subtraction Formulas

Corollaries:

(1) For any values of A and B,
(i) sin (A + B) . sin (A – B) = sin2 A – sin2 B
(ii) cos (A + B) . cos (A – B) = cos2 A – sin2 B

Proof: (i) sin (A + B) . sin (A – B) = (sin A cos B + cos A sin B) (sin A cos B – cos A sin B) = sin2 A cos2 B – cos2 A sin2 B = sin2 A (1- sin2 B) – (1 – sin2 A) sin2 B = sin A – sin2 A sin2 B – sin2 B + sin2 A sin2 B = sin2 A – sin2 B

(ii) cos (A + B) . cos (A – B) = (cos A cos B – sin A sin B) (cos A cos B + sin A sin B) = cos2 A cos2 B – sin2 A sin2 B = cos2 A (1 – sin2 B) – (1 – cos2 A) sin2 B = cos2 A – cos2 A sin2 B – sin2 B + cos2 A sin2 B = cos2 A – sin2 B
(2) For any values of A, B and C,
(i) sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C – tan A tan B tan C)
(ii) cos (A + B + C) = cos A cos B cos C (1- tan A tan B – tan B tan C – tan C tan A)

Proof: (i) sin (A + B + C) = sin {A + (B + C)} = sin A cos (B + C) + cos A sin (B + C) = sin A (cos B cos C – sin B sin C) + cos A (sin B cos C + cos B sin C) = cos A cos B cos C [(sin A cos B cos C – sin A sin B sin C + cos A sin B cos C + cos A cos B sin C)/cosA cosB cos C] = cos A cos B cos C [tan A – tan A tan B tan C + tan B + tan C] = cos A cos B cos C [tan A + tan B + tan C – tan A tan B tan C]

(ii) cos (A + B + C) = cos {A + (B + C)} = cos A cos (B + C) – sin A sin (B + C) = cos A (cos B cos C – sin B sin C) – sin A (sin B cos C + cos B sin C) = cos A cos B cos C – cos A sin B sin C – sin A sin B cos C – sin A cos B sin C = cos A cos B cos C [1 – (cos A sin B sin C + sin A sin B sin C + sin A cos B sin C)/cos A cos B cos C] = cos A cos B cos C [1 – tan B tan C – tan A tan B – tan C tan A] = cos A cos B cos C [1- tan A tan B – tan B tan C – tan C tan A]

(3) For any values of A, B and C, tan (A + B + C) = (tan A + tan B + tan C – tan A tan B tan C)/(1- tan A tan B – tan B tan C – tan C tan A)

Proof:

Trigonometry formula proof
Important Formulas:

(1) sin (A + B) = sin A cos B + cos A sin B
(2) sin (A – B) = sin A cos B – cos A sin B
(3) cos (A + B) = cos A cos B – sin A sin B
(4) cos (A – B) = cos A cos B + sin A sin B
(5) tan (A + B) = (tan A + tan B)/(1 – tan A tan B), tan A tan B ≠ 1
(6) tan (A – B) = (tan A – tan B)/(1 + tan A tan B)
(7) cot (A + B) = (cot A cot B – 1)/(cot B + cot A)
(8) cot (A – B) = (cot A cot B + 1)/(cot B – cot A)
(9) sin (A + B). sin (A- B) = sin2 A – sin2 B
(10) cos (A + B). cos (A- B) = cos2 A – sin2 B = cos2 B – sin2 A
(11) sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C – tan A tan B tan C)
(12) cos (A + B + C) = cos A cos B cos C (1- tan A tan B – tan B tan C – tan C tan A)
(13) tan (A + B + C) = (tan A + tan B + tan C – tan A tan B tan C)/(1- tan A tan B – tan B tan C – tan C tan A)

Periodic Function in Trigonometry
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Primary Cells- Dry Cell and Mercury Cell
Applications of Photoelectric Effect
Heat Transfer and Solar Energy– NIOS

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