Triple angle formulas 1.jpg

These formulas can be derived using x + y formulas

 

For sin 3x

sin 3x = sin (2x + x)

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

   = sin 2x cos x + sin x cos 2x

Using cos 2x = 1 – 2 sin 2 x and sin 2x = 2 sin x cos x

   = (2 sin x cos x) cos x + sin x (1 – 2sin 2 x)

   = 2 sin x cos 2 x + sin x – 2sin 3 x

Using cos 2 x = 1 – sin 2 x

   = 2 sin x (1 – sin 2 x) + sin x – 2sin 3 x)

   = 2 sin x – 2sin 3 x + sin x – 2sin 3 x

   = 3 sin x – 4 sin 3 x

 

For cos 3 x

cos 3x = cos (2x + x)

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

   = cos 2x cos x – sin 2x sin x

Using cos 2x = 2 cos 2 x – 1 and sin 2x = 2 sin x cos x

   = (2cos 2 x – 1) cos x – (2 sin x cos x) sin x

   = 2cos 2 x – cos x – 2 sin 2 x cos x

Using sin 2 x = 1 – cos 2 x

   = 2cos 3 x – cos x – 2 (1 – cos 2 x) cos x

   = 2cos 3 x – cos x – 2 cos x (1 – cos 2 x)

   = 2cos 3 x – cos x – 2 cos x + 2 cos 3 x

   = 4cos 3 x – 3cos x

 

For tan 3x

Triple angle formulas - Part 2

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Transcript

These formulas can be derived using x + y formulas For sin 3x sin 3x = sin (2x + x) Using sin (x + y) = sin x cos y + cos x sin y = sin 2x cos x + sin x cos 2x Using cos 2x = 1 – 2 sin2 x and sin 2x = 2 sin x cos x = (2 sin x cos x) cos x + sin x (1 – 2sin2 x) = 2 sin x cos2 x + sin x – 2sin3 x Using cos2 x = 1 – sin2 x = 2 sin x (1 – sin2 x) + sin x – 2sin3 x) = 2 sin x – 2sin3 x + sin x – 2sin3 x = 3 sin x – 4 sin3 x For cos 3x cos 3x = cos (2x + x) Using cos (x + y) = cos x cos y – sin x sin y = cos 2x cos x – sin 2x sin x Using cos 2x = 2 cos2 x – 1 and sin 2x = 2 sin x cos x = (2cos2 x – 1) cos x – (2 sin x cos x) sin x = 2cos2 x – cos x – 2 sin2 x cos x Using sin2 x = 1 – cos2 x = 2cos2 x – cos x – 2 (1 – cos2 x) cos x = 4cos2 x – 3cos x For tan 3x

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.