Double angle formulas.jpg

These formulas can be derived using x + y formulas

For sin 2x

sin 2x = sin (x + x)

Using sin (x + y) = sin x cos y + cos x sin y

   = sin x cos x + sin x cos x

   = 2 sin x cos x

 

For cos 2x

cos 2x = cos (x + x)

Using cos (x + y) = cos x cos y – sin x sin y

   = cos x cos x – sin x sin x

   = cos 2 x – sin 2 x

 

Putting cos 2 x = 1 – sin 2 x

cos 2x = cos 2 x – sin 2 x

     = 1 – sin 2 x – sin 2 x

     = 1 – 2sin 2 x

 

Putting sin 2 x = 1 – cos 2 x

cos 2x = cos 2 x – sin 2 x

     = cos 2 x – (1 – cos 2 x)

     = cos 2 x – 1 + cos 2 x

     = 2cos 2 x – 1

 

For tan 2x

Double angle formulas - Part 2

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Transcript

These formulas can be derived using x + y formulas For sin 2x sin 2x = sin (x + x) Using sin (x + y) = sin x cos y + cos x sin y = sin x cos x + sin x cos x = 2 sin x cos x For cos 2x cos 2x = cos (x + x) Using cos (x + y) = cos x cos y – sin x sin y = cos x cos x – sin x sin x = cos2 x – sin2 x Putting cos2 x = 1 – sin2 x cos 2x = cos2 x – sin2 x = 1 – sin2 x – sin2 x = 1 – 2sin2 x Putting sin2 x = 1 – cos2 x cos 2x = cos2 x – sin2 x = cos2 x – (1 – cos2 x) = cos2 x – 1 + cos2 x = 2cos2 x – 1 For tan 2x

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.