Trigonometry: Basic Concepts, Formulas

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Trigonometric Identities and Formulas

There are three systems of measuring Angles-

1. Circular System

  • Unit of Measurement is radian.
  • 180 degree = π radians

2. Sexagesimal System (English System)

  • Right angle is divided into 90 equal parts called degree.
  • Unit of Measurement is degree.
  • Each degree is divided into 60 equal parts called minute. (1 degree = 60’)
  • Each minute divided into 60 equal parts called seconds (1 minute = 60’’)

3. Centesimal or French System

  • Right angle is divided into 100 equal parts.
  • Unit of measurement is grades.
  • Each grade is divided into 100 equal parts called minute, and minutes into seconds.

Sign Conventions

  • cos (90 – θ) = sinθ
  • tan (90 – θ) = cotθ
  • cosec (90 – θ) = secθ
  • sec (90 – θ) = cosecθ
  • cot (90 – θ) = tanθ

Some Basic Formulas and Identities

\sin ^{ 2 }{ \theta }+\cos ^{ 2 }{ \theta } =1 1+\tan ^{ 2 }{ \theta } =\sec ^{ 2 }{ \theta } 1+\cot ^{ 2 }{ \theta } =\csc ^{ 2 }{ \theta }

Advanced Trigonometric Identities

sin (A + B) = sin A × cos B + cos A × sin B

sin (A – B) = sin A × cos B – cos A × sin B

cos (A + B) = cos A × cos B – sin A × sin B

cos (A – B) = cos A × cos B + sin A × sin B

tan (A + B) = (tan A + tan B) / (1 – tan A tan B)

tan (A – B) = (tan A – tan B) / (1+ tan A tan B)

cot (A + B) = (cot A cot B – 1) / (cot A + cot B)

cot (A – B) = (cot A cot B + 1) / (cot B – cot A)

sin 2θ = 2sinθcosθ

2sin A sin B = cos (A – B) – cos (A + B)

2cos A cos B = cos (A + B) + cos (A – B)

2sin A cos B = sin (A + B) + sin (A – B)

2cos A sin B = sin (A + B) – sin (A – B)

sinC + sinD = 2sin[(C + D)/2] × cos[(C – D)/2]

sinC – sinD = 2cos[(C + D)/2] × sin[(C – D)/2]

cosC + cosD = 2cos[(C + D)/2]cos[(C – D)/2]

cosC – cosD = 2sin[(C + D)/2]cos[(D – C)/2]

\tan { (A+B+C) } =\frac { \tan { A } +\tan { B } +\tan { C } -(\tan { A } \tan { B } \tan { C } ) }{ 1-\tan { A } \tan { B } -\tan { B } \tan { C } -\tan { C } \tan { A } }

\sin{ 2\theta =\frac{2\tan{2\theta}}{1+\tan ^{ 2 }{\theta}}}

\cos{2\theta =2\cos ^{ 2 }{\theta-1 }}

\cos{2\theta=\cos^{2}{\theta-\sin ^{2}{\theta}}}

\cos { 2\theta =1-2\sin ^{ 2 }{ \theta}}

\cos { 2\theta } =\frac { 1-\tan ^{ 2 }{ \theta } }{ 1+\tan ^{ 2 }{ \theta } }

\tan { 2\theta} =\frac { 2\tan{\theta }}{1-\tan ^{ 2 }{\theta}}

\sin { 3\theta } =3\sin { \theta } -4\sin ^{ 3 }{ \theta }

\cos {3\theta}=4\cos ^{ 3 }{ \theta} -3\cos {\theta}

\tan { 3\theta }=\frac { 3\tan { \theta } -\tan ^{ 3 }{ \theta } }{ 1-3\tan ^{ 2 }{ \theta } }