Trigonometric Formulae, Study notes of Trigonometry

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𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 = 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 − 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 − 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒊𝒊𝒊𝒊𝒕𝒕 = 𝒕𝒕 + 𝒕𝒕

𝒕𝒕

𝒕𝒕

Values of trigonometric functions

(1) sin( A + B + C ) = sin A cos B cos C + cos A sin B cos C +cos A cos B sin C − sin A sin B sin C (2) cos( A + B + C ) = cos A cos B cos C − sin A sin B cos C −sin A cos B sin C − cos A sin B sin C (3) A B B C C A A B C A B C A B C 1 tan tan tan tan tan tan tan tan tan tan tan tan tan( ) − − −

• • −
• • = Formulae for the trigonometric ratios of sum and differences of three angles

imp results 1 ] 𝒄𝒄𝒊𝒊𝒕𝒕�𝒄𝒄𝟎𝟎 𝟎𝟎 − 𝜽𝜽�𝒄𝒄𝒊𝒊𝒕𝒕𝜽𝜽 𝒄𝒄𝒊𝒊𝒕𝒕�𝒄𝒄𝟎𝟎 𝟎𝟎

• 𝜽𝜽� = 𝟏𝟏 𝟒𝟒 𝒄𝒄𝒊𝒊𝒕𝒕𝟑𝟑𝜽𝜽 2 ] 𝒕𝒕𝒄𝒄𝒄𝒄�𝒄𝒄𝟎𝟎 𝟎𝟎 − 𝜽𝜽�𝒕𝒕𝒄𝒄𝒄𝒄𝜽𝜽 𝒕𝒕𝒄𝒄𝒄𝒄�𝒄𝒄𝟎𝟎 𝟎𝟎
• 𝜽𝜽� = 𝟏𝟏 𝟒𝟒 𝒕𝒕𝒄𝒄𝒄𝒄𝟑𝟑𝜽𝜽 3 ] 𝒕𝒕𝒕𝒕𝒕𝒕�𝒄𝒄𝟎𝟎 𝟎𝟎 − 𝜽𝜽�𝒕𝒕𝒕𝒕𝒕𝒕𝜽𝜽 𝒕𝒕𝒕𝒕𝒕𝒕�𝒄𝒄𝟎𝟎 𝟎𝟎
• 𝜽𝜽� = 𝒕𝒕𝒕𝒕𝒕𝒕𝟑𝟑𝜽𝜽

IMP RESULTS

1] For ∆𝒕𝒕𝒕𝒕𝒕𝒕, 𝒄𝒄𝒊𝒊𝒕𝒕𝒄𝒄𝒕𝒕 + 𝒄𝒄𝒊𝒊𝒕𝒕𝒄𝒄𝒕𝒕 + 𝒄𝒄𝒊𝒊𝒕𝒕𝒄𝒄𝒕𝒕 = 𝟒𝟒𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 2 ] For ∆𝒕𝒕𝒕𝒕𝒕𝒕, 𝒕𝒕𝒄𝒄𝒄𝒄𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒄𝒄𝒄𝒄𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒄𝒄𝒄𝒄𝒄𝒄𝒕𝒕 = 𝟏𝟏 − 𝟒𝟒𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 3 ] For ∆𝒕𝒕𝒕𝒕𝒕𝒕, 𝒕𝒕𝒕𝒕𝒕𝒕𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒕𝒕𝒕𝒕𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒕𝒕𝒕𝒕𝒄𝒄𝒕𝒕 = 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝐭𝐭𝐭𝐭𝐭𝐭 𝒕𝒕 4 ] For ∆𝒕𝒕𝒕𝒕𝒕𝒕, 𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 + 𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 + 𝒄𝒄𝒊𝒊𝒕𝒕𝒕𝒕 = 𝟒𝟒𝒕𝒕𝒄𝒄𝒄𝒄 𝒕𝒕 𝒄𝒄

𝒕𝒕 𝒄𝒄

𝒕𝒕 𝒄𝒄 5 ] For ∆𝒕𝒕𝒕𝒕𝒕𝒕, 𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 + 𝒕𝒕𝒄𝒄𝒄𝒄𝒕𝒕 = 𝟏𝟏 + 𝟒𝟒𝒄𝒄𝒊𝒊𝒕𝒕 𝒕𝒕 𝒄𝒄

𝒕𝒕 𝒄𝒄

𝒕𝒕 𝒄𝒄 6 ] For ∆𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 + 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 + 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 = 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝐭𝐭𝐭𝐭𝐭𝐭 𝒕𝒕