Reduce R=4Sen³x.Cosx+4Cos³x.Senx
Answers
Answer:
Step-by-step explanation:
cos(3x) = 4cos^3(x) − 3cos(x)
Here are some Important results.
Pythagorean Identities
sin 2X + cos 2X = 1
1 + tan 2X = sec 2X
1 + cot 2X = csc 2X
Negative Angle Identities
sin (-X) = – sin X, odd function
csc (-X) = – csc X, odd function
cos (-X) = cos X, even function
sec (-X) = sec X, even function
tan (-X) = – tan X, odd function
cot (-X) = – cot X, odd function
Cofunctions Identities
sin (π /2 – X) = cos X
cos (π /2 – X) = sin X
tan (π /2 – X) = cot X
cot (π/2 – X) = tan X
sec (π /2 – X) = csc X
csc (π /2 – X) = sec X
Addition Formulas
cos (X + Y) = cos X cos Y – sin X sin Y
cos (X – Y) = cos X cos Y + sin X sin Y
sin (X + Y) = sin X cos Y + cos X sin Y
sin (X – Y) = sin X cosY – cos X sin Y
tan (X + Y) = [ tan X + tan Y ] / [ 1 – tan X tan Y]
tan (X – Y) = [ tan X – tan Y ] / [ 1 + tan X tan Y]
cot (X + Y) = [ cot X cot Y – 1 ] / [ cot X + cot Y]
cot (X – Y) = [ cot X cot Y + 1 ] / [ cot Y – cot X]
Sum to Product Formulas
cos X + cos Y = 2cos[(X + Y)/ 2] cos[(X – Y)/ 2]
sin X + sin Y = 2sin[(X + Y)/ 2] cos[(X – Y)/ 2]
Difference to Product Formulas
cos X – cos Y = – 2sin[(X + Y) / 2] sin[(X – Y) / 2]
sin X – sin Y = 2cos[(X + Y) / 2] sin[(X – Y) / 2]
Product to Sum/Difference Formulas
cos X cos Y = (1/2) [cos (X – Y) + cos (X + Y)]
sin X cos Y = (1/2) [sin (X + Y) + sin (X – Y)]
cos X sin Y = (1/2) [sin (X + Y) – sin[ (X – Y)]
sin X sin Y = (1/2) [cos (X – Y) – cos (X + Y)]
Difference of Squares Formulas
sin 2X – sin 2Y = sin (X + Y) sin (X – Y)
cos 2X – cos 2Y = – sin (X + Y) sin (X – Y)
cos 2X – sin 2Y = cos (X + Y) cos (X – Y)
Double Angle Formulas
sin (2X) = 2 sin X cos X
cos (2X) = 1 – 2sin 2X = 2cos 2X – 1
tan (2X) = 2tan X/[1 – tan 2X]
Multiple Angle Formulas
sin (3X) = 3sin X – 4sin 3X
cos (3X) = 4cos 3X – 3cos X
sin (4X) = 4sin X cos X – 8sin 3X cos X
cos (4X) = 8cos 4X – 8cos 2X + 1
Half Angle Formulas
sin (X/2) = ±√[(1 – cos X)/2]
cos (X/2) = ±√[(1 + cos X)/2]
tan (X/2) = ±√[(1 – cos X)/(1 + cos X)]
= sin X/(1 + cos X)
= (1 – cos X)/sin X
Power Reducing Formulas
sin 2X = 1/2 – (1/2) cos (2X))
cos 2X = 1/2 + (1/2) cos (2X))
sin 3X = (3/4) sin X – (1/4) sin (3X)
cos 3X = (3/4) cos X + (1/4) cos (3X)
sin 4X = (3/8) – (1/2)cos (2X) + (1/8)cos (4X)
cos 4X = (3/8) + (1/2)cos (2X) + (1/8)cos (4X)
sin 5X = (5/8)sin X – (5/16)sin (3X) + (1/16)sin (5X)
cos 5X = (5/8)cos X + (5/16)cos (3X) + (1/16)cos (5X)
sin 6X = 5/16 – (15/32)cos (2X) + (6/32)cos (4X) – (1/32)cos (6X)
cos 6X = 5/16 + (15/32)cos (2X) + (6/32)cos (4X) + (1/32)cos (6X)
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Trigonometric Functions Periodicity
sin (X + 2π) = sin X, period 2π
cos (X + 2π) = cos X, period 2π
sec (X + 2π) = sec X, period 2π
csc (X + 2π) = csc X, period 2π
tan (X + π) = tan X, period π
cot (X + π) = cot X, period π
Hope this will help