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 } }