Half Angle Formulas Derivation Using Double Angle Formulas

Half-Angle formulas are derived using double-angle formulas. Before learning about half-angle formulas we must learn about Double-angle in Trigonometry, most commonly used double-angle formulas in trigonometry are:

sin 2x = 2 sin x cos x
cos 2x = cos² x – sin² x
= 1 – 2 sin²x
= 2 cos²x – 1
tan 2x = 2 tan x / (1 – tan²x)

Now replacing x with x/2 on both sides in the above formulas we get

sin x = 2 sin(x/2) cos(x/2)
cos x = cos² (x/2) – sin² (x/2)
= 1 – 2 sin² (x/2)
= 2 cos²(x/2) – 1
tan A = 2 tan (x/2) / [1 – tan²(x/2)]

Half-Angle Formula for Cos Derivation

We use cos2x = 2cos²x – 1 for finding the Half-Angle Formula for Cos

Put x = 2y in the above formula

cos (2)(y/2) = 2cos²(y/2) – 1

cos y = 2cos²(y/2) – 1

1 + cos y = 2cos²(y/2)

2cos²(y/2) = 1 + cosy

cos²(y/2) = (1+ cosy)/2

cos(y/2) = ± √{(1+ cosy)/2}

Half-Angle Formula for Sin Derivation

We use cos 2x = 1 – 2sin²x for finding the Half-Angle Formula for Sin

Put x = 2y in the above formula

cos (2)(y/2) = 1 – 2sin²(y/2)

cos y = 1 – 2sin²(y/2)

2sin²(y/2) = 1 – cosy

sin²(y/2) = (1 – cosy)/2

sin(y/2) = ± √{(1 – cosy)/2}

Half-Angle Formula for Tan Derivation

We know that tan x = sin x / cos x such that,

tan(x/2) = sin(x/2) / cos(x/2)

Putting the values of half angle for sin and cos. We get,

tan(x/2) = ± [(√(1 – cosy)/2 ) / (√(1+ cosy)/2 )]

tan(x/2) = ± [√(1 – cosy)/(1+ cosy) ]

Rationalising the denominator

tan(x/2) = ± (√(1 – cosy)(1 – cosy)/(1+ cosy)(1 – cosy))

tan(x/2) = ± (√(1 – cosy)²/(1 – cos²y))

tan(x/2) = ± [√{(1 – cosy)²/( sin²y)}]

tan(x/2) = (1 – cosy)/( siny)

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