Tuesday, May 20, 2008

Revision Material Ch 33. Trigonometric Ratios - 2

sin (A+B+C) = sin A cos B cos C + sin B cos A cos C + sin C cos A cos B - sin A sin B sin C

sin 3A = 3 sin A - 4 sin³A

Cos (A+B+C) = cos A cos B cos C - cos A sin B sin C - Cos B sin A sin C - cos C sin A sin B

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]

Transformation Formulae

sin (A+B) + sin (A-B) = 2 sin A cos B

2 sin A cos B = sin (A+B) + sin (A-B)

sin (A+B) - sin (A-B) = 2cos A sin B

2cos A sin B = sin (A+B) - sin (A-B)

cos (A+B) + cos (A-B) = 2cos A cos B

2cos A cos B = cos (A+B) + cos (A-B)

cos (A+B) - cos (A-B) = 2sin A sin B

2sin A sin B = cos (A+B) - cos (A-B)

Therefore

2 sin A cos B = sin (A+B) + sin (A-B)
2cos A sin B = sin (A+B) - sin (A-B)
2cos A cos B = cos (A+B) + cos (A-B)
2sin A sin B = cos (A+B) - cos (A-B)

The products of two sines or two cosines and one sine and one cosine can be transformed into the sum or differences of two sines or two cosines.

Trigonometric ratios of multiple angles

Trigonometric ratios of angle 2A in terms of an angle A

Sin 2A = 2sin A cos A
Sin 2 A = 2tan A/(1 + tan² A)

Cos 2A = cos² A - sin² A
Cos 2A = 2cos² A – 1
Cos 2A = 1 – 2sin² A
cos 2A = (1- tan² A)/(1+ tan² A)

tan 2A = 2tan A/(1 - tan² A)

Trigonometric ratios of angle 3A in terms of an angle A

Sin 3A = 3sin A - sin³ A

cos 3A = 4cos³ A – 3cos A

tan 3A = (3tan A - tan³ A)/(1-3tan² A)

Trigonometric ratios of sub-multiple angles

Trigonometric ratios of angle A in terms of an angle A/2

sin A = 2sin A/2 cos A/2
sin A = (2tan A/2)/(1 + tan² A/2)

cos A = cos² A/2 - sin² A/2
cos A = 2cos² A/2 – 1
cos A = 1 – 2sin² A/2
cos A = (1- tan² A/2)/(1+ tan² A/2)

tan A = (2tan A/2)/(1 - tan² A/2)

Trigonometric ratios of angle A in terms of an angle A/3

Sin A = 3sin (A/3) - sin³ (A/3)

cos A = 4cos³ (A/3) – 3cos (A/3)

tan A = (3tan (A/3) - tan³ (A/3))/(1-3tan² (A/3))

Trigonometric ratios of angle A/2 in terms of an angle cos A

cos A/2 = ±√[(1+cos A)/2]

sin A/2 = ±√[(1-cos A)/2]

tan A/2 = ±√[(1-cos A)/(1 + cos a)]