Trigonometric Formulas:
Below are the list of all trigonometric functions formulas;
Basic Formulas:
` \ sin \ left( \theta\right) \ = \ \ frac1 {\ csc( \theta)}\ `
OR
` \ csc \ left( \theta\right)\ = \ \ frac1 {\ sin\ left( \theta\right)}`
=====
` \ cos \ left( \theta\right) \ = \ \ frac1 {\ sec( \theta)}\ `
OR
` sec \ left( \theta\right) \ = \ \ frac1 {\ cos\ left( \theta\right)} `
=====
` \ tan\ left( \theta\right) \ = \ \frac {\ sin\ left( \theta\right)}{\ cos\ left( \theta\right)}\ `
OR
` \ tan\ left( \theta\right) \ = \ \frac1 {\ cot\ left( \theta\right)} `
=====
` \ cot\ left( \theta\right)\ = \ \ frac {\ cos\ left( \theta\right)}{\ sin \ left( \theta\right)}`
OR
` \ cot \ left( \theta\right) \ = \ \ frac1 {\ tan\ left( \theta\right)} `
=====
` \ sin^2\ left( \theta\right) \ + \ \ cos^2 \ left( \theta\right)\ = \ 1 \ `
OR
` \ sin^2\ left( \theta\right) \ = \ 1 \ - \ \ cos^2 \ left( \theta\right) `
OR
` \ cos^2 \ left( \theta\right) \ = \ 1 \ - \ \ sin^2 \ left( \theta\right) `
=====
` 1 \ + \ \ tan^2 \ left( \theta\right) \ = \ \ sec^2 \ left( \theta\right) `
OR
` \ tan^2 \ left( \theta\right) \ = \ \ sec^2 \ left(\theta\right) \ - \ 1 `
=====
` 1 \ + \ \cot^2\ left( \theta\right) \ = \ \ csc^2 \ left(\theta\right) `
OR
` \ cot^2 \ left( \theta\right) \ = \ \ csc^2 \ left( \theta\right) \ - \ 1 `
=====
` \ csc \ left( \theta\right)\ = \ \ frac1 {\ sin\ left( \theta\right)}`
=====
` sec \ left( \theta\right) \ = \ \ frac1 {\ cos\ left( \theta\right)} `
=====
=====
NEGATIVE(-ve) ANGLE FORMULAS:
` \ sin \ left( -\theta\right) \ =\ - \ sin\ left( \theta\right) `
` \ cos \ left( -\theta\right) \ = \ \ cos\ left( \theta\right) `
` \ tan \ left( -\theta\right) \ = \ - \ tan\ left( \theta\right) `
` \ cot \ left( -\theta\right) \ = \ - \ cot\ left( \theta\right) `
` \ sec \ left( -\theta\right) \ =\ \ sec\ left( \theta\right) `
` \ csc \ left( -\theta\right) \ = \ - \ csc\ left( \theta\right) `
` \ sin \ left( -\theta\right) \ =\ - \ sin\ left( \theta\right) `
` \ cos \ left( -\theta\right) \ = \ \ cos\ left( \theta\right) `
` \ tan \ left( -\theta\right) \ = \ - \ tan\ left( \theta\right) `
` \ cot \ left( -\theta\right) \ = \ - \ cot\ left( \theta\right) `
` \ sec \ left( -\theta\right) \ =\ \ sec\ left( \theta\right) `
` \ csc \ left( -\theta\right) \ = \ - \ csc\ left( \theta\right) `
` \ cos \ left( -\theta\right) \ = \ \ cos\ left( \theta\right) `
` \ tan \ left( -\theta\right) \ = \ - \ tan\ left( \theta\right) `
` \ cot \ left( -\theta\right) \ = \ - \ cot\ left( \theta\right) `
` \ sec \ left( -\theta\right) \ =\ \ sec\ left( \theta\right) `
` \ csc \ left( -\theta\right) \ = \ - \ csc\ left( \theta\right) `
DOUBLE ANGLE FORMULAS:
` \ sin \ left( 2\theta\right) \ = \ 2 \ sin\ left( \theta\right) \ cos\ left( \theta\right) `
OR
` \ sin \ left( 2\theta\right)\ = \ \ frac { 2 \tan\left( \theta\right)}{ 1\ + \ \ tan^2\ left( \theta\right)} `
=====
` \ cos \ left( 2\theta\right) \ = \ \ cos^2\ left( \theta\right)\ -\ \ sin^2\ left( \theta\right) `
OR
` \ cos \ left( 2\theta\right) \ = \ 2 \ cos^2 \ left( \theta\right)\ - \ 1 `
OR
` \ cos \ left( 2\theta\right) \ = \ 1\ - \ 2 \ sin^2\ left( \theta\right) `
OR
` \ cos \ left( 2\theta\right) \ = \ \ frac {1\ - \ \ tan^2\ left( \theta\right)}{1\ + \ \ tan^2\ left( \theta\right)} `
= = = = =
` \ tan \ left( 2\theta\right) \ = \ \ frac { 2 \ tan \ left( \theta\right)}{ 1-\ tan^2\ left( \theta\right)} `
` \ sin \ left( 2\theta\right) \ = \ 2 \ sin\ left( \theta\right) \ cos\ left( \theta\right) `
OR
` \ sin \ left( 2\theta\right)\ = \ \ frac { 2 \tan\left( \theta\right)}{ 1\ + \ \ tan^2\ left( \theta\right)} `
=====
` \ cos \ left( 2\theta\right) \ = \ \ cos^2\ left( \theta\right)\ -\ \ sin^2\ left( \theta\right) `
OR
` \ cos \ left( 2\theta\right) \ = \ 2 \ cos^2 \ left( \theta\right)\ - \ 1 `
OR
` \ cos \ left( 2\theta\right) \ = \ 1\ - \ 2 \ sin^2\ left( \theta\right) `
OR
` \ cos \ left( 2\theta\right) \ = \ \ frac {1\ - \ \ tan^2\ left( \theta\right)}{1\ + \ \ tan^2\ left( \theta\right)} `
= = = = =
` \ tan \ left( 2\theta\right) \ = \ \ frac { 2 \ tan \ left( \theta\right)}{ 1-\ tan^2\ left( \theta\right)} `
HALF ANGLE FORMULAS:
` \ sin \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1- \ cos\ left( \ alpha \ right)} 2} `
` \ cos \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1+ \ cos\ left( \ alpha \ right)} 2} `
` \
tan \ left( \ frac \alpha2\ right) \ = \ pm \ sqrt{ \ frac{1 - \ cos\
left( \ alpha \ right)} {1 + \ cos\ left( \ alpha \ right)}} `
` \ sin \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1- \ cos\ left( \ alpha \ right)} 2} `
` \ cos \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1+ \ cos\ left( \ alpha \ right)} 2} `
` \
tan \ left( \ frac \alpha2\ right) \ = \ pm \ sqrt{ \ frac{1 - \ cos\
left( \ alpha \ right)} {1 + \ cos\ left( \ alpha \ right)}} `
FUNDAMENTAL LAWS:
`
\ sin \ left( \ alpha + \ beta \ right) = \ sin\ left( \ alpha \ right)
\ cos\ left( \ beta \ right) + \ cos\ left( \ alpha \ right)\ sin\
left(\ beta \ right) `
`
\ sin\ left( \ alpha - \ beta \ right) = \ sin\ left( \ alpha \ right) \
cos\ left( \ beta \ right) - \ cos\ left( \ alpha \ right)\ sin\ left(\
beta \ right) `
`
\ cos\ left( \ alpha + \ beta \ right) = \ cos\ left( \ alpha \ right) \
cos\ left( \ beta \ right) - \ sin\ left( \ alpha \ right)\ sin\ left(\
beta \ right) `
`
\ cos\ left( \ alpha - \ beta \ right) = \ cos\ left( \ alpha \ right) \
cos\ left( \ beta \ right) + \ sin\ left( \ alpha \ right)\ sin\ left(\
beta \ right) `
`
\ tan\ left( \ alpha + \ beta \ right) = \ frac { \ tan \ left( \
alpha \ right) + \ tan\ left( \ beta \ right) } {1- \ tan\ left( \ alpha
\ right) \ tan \ left( \ beta \ right) }`
`\
tan\ left( \ alpha -\ beta \ right) = \ frac { \ tan \ left( \ alpha \
right) -\ tan \ left( \ beta \ right) } {1 + \ tan\ left( \ alpha \
right) \ tan \ left( \ beta \ right) } `
` \
cot \ left( \ alpha + \ beta \ right) = \ frac { \ cot \ left( \ alpha \
right) \ cot\ left( \ beta \ right) - 1} { \ cot\ left( \ alpha \
right) + \ cot \ left( \ beta \ right)} `
` \
cot \ left( \ alpha -\ beta \ right) = \ frac { \ cot \ left( \ alpha
\right) \ cot\ left( \ beta \ right) + 1} { \ cot\ left( \ beta \ right)
- \ cot \ left( \ alpha \ right)} `
PRODUCT TO SUM FORMULAS:
`
\ sin \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \
frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) + \ sin \ left( \
alpha - \ beta \ right) \ right] `
`
\ cos \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = \
frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) - \ sin \ left( \
alpha - \ beta \ right) \ right] `
`
\ cos \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \
frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) + \ cos \ left( \
alpha - \ beta \ right) \ right] `
`
\ sin \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = - \
frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) - \ cos \ left( \
alpha - \ beta \ right) \ right] `
`
\ sin \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \
frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) + \ sin \ left( \
alpha - \ beta \ right) \ right] `
`
\ cos \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = \
frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) - \ sin \ left( \
alpha - \ beta \ right) \ right] `
`
\ cos \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \
frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) + \ cos \ left( \
alpha - \ beta \ right) \ right] `
`
\ sin \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = - \
frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) - \ cos \ left( \
alpha - \ beta \ right) \ right] `
SUM TO PRODUCT FORMULAS:
` \
sin \ left( \ upsilon \ right) + \ sin \ left( \ nu \ right) = 2 \ sin \
left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \
upsilon - \ nu} 2 \ right) `
` \ sin \ left( \
upsilon \ right) -\ sin \ left( \ nu \ right) = 2 \ cos \ left( \ frac {
\ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2
\ right) `
` \ cos \ left( \ upsilon \ right) +
\ cos \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \
nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right) `
` \
cos \ left( \ upsilon \ right) -\ cos \ left( \ nu \ right) = - 2 \ sin
\ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \
upsilon - \ nu} 2 \ right) `
` \
sin \ left( \ upsilon \ right) + \ sin \ left( \ nu \ right) = 2 \ sin \
left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \
upsilon - \ nu} 2 \ right) `
` \ sin \ left( \
upsilon \ right) -\ sin \ left( \ nu \ right) = 2 \ cos \ left( \ frac {
\ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2
\ right) `
` \ cos \ left( \ upsilon \ right) +
\ cos \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \
nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right) `
` \
cos \ left( \ upsilon \ right) -\ cos \ left( \ nu \ right) = - 2 \ sin
\ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \
upsilon - \ nu} 2 \ right) `
LAW OF SINE:
` \
frac { \ sin \ left( \ alpha \ right) } a = \ frac{ \ sin \ left( \
beta \ right) } b = \ frac{ \ sin \ left( \ gamma \ right) } c `
OR
` \
frac a { \ sin \ left( \ alpha \ right) } = \ frac b { \sin \ left( \
beta \ right) } = \ frac c { \ sin \ left( \ gamma \ right) } `
OR
` a
\ : \ b \ : \ c \ = \ \ sin\ left( \ alpha \ right) \ : \ \ sin \
left( \ beta \ right) \ : \ \ sin \ left( \ gamma \ right) `
` \
frac { \ sin \ left( \ alpha \ right) } a = \ frac{ \ sin \ left( \
beta \ right) } b = \ frac{ \ sin \ left( \ gamma \ right) } c `
OR
` \
frac a { \ sin \ left( \ alpha \ right) } = \ frac b { \sin \ left( \
beta \ right) } = \ frac c { \ sin \ left( \ gamma \ right) } `
OR
` a
\ : \ b \ : \ c \ = \ \ sin\ left( \ alpha \ right) \ : \ \ sin \
left( \ beta \ right) \ : \ \ sin \ left( \ gamma \ right) `
LAW OF COSINE:
` \ cos \ left( \ alpha \ right) = \ frac { b^2 + c^2 - a^2} { 2bc } `
OR
` a^2 = b^2 + c^2 -2bc \ cos \ left( \ alpha \ right) `
=====
` \ cos \ left( \ beta \ right) = \ frac { a^2 + c^2 - b^2} { 2ac } `
OR
` b^2 = a^2 + c^2 -2ac \ cos \ left( \ beta \ right) `
=====
` \ cos \ left( \ gamma \ right) = \ frac { a^2 + b^2 - c^2} { 2ab } `
OR
` c^2 = a^2 + b^2 -2ab \ cos \ left( \ gamma \ right) `
` \ cos \ left( \ alpha \ right) = \ frac { b^2 + c^2 - a^2} { 2bc } `
OR
` a^2 = b^2 + c^2 -2bc \ cos \ left( \ alpha \ right) `
=====
` \ cos \ left( \ beta \ right) = \ frac { a^2 + c^2 - b^2} { 2ac } `
OR
` b^2 = a^2 + c^2 -2ac \ cos \ left( \ beta \ right) `
=====
` \ cos \ left( \ gamma \ right) = \ frac { a^2 + b^2 - c^2} { 2ab } `
OR
` c^2 = a^2 + b^2 -2ab \ cos \ left( \ gamma \ right) `
HALF ANGLE FORMULAS:
` \ sin \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { bc } }`
` \ sin \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { ac } }`
` \ sin \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { ab } }`
` \ cos \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { s (s-a) } { bc } }`
` \ cos \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { s (s-b) } { ac } }`
` \ cos \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { s (s-c) } { ab } }`
` \ tan \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { s (s-a) } } `
` \ tan \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { s (s-b) } } `
` \ tan \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { s (s-c) } } `
` \ sin \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { bc } }`
` \ sin \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { ac } }`
` \ sin \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { ab } }`
` \ cos \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { s (s-a) } { bc } }`
` \ cos \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { s (s-b) } { ac } }`
` \ cos \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { s (s-c) } { ab } }`
` \ tan \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { s (s-a) } } `
` \ tan \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { s (s-b) } } `
` \ tan \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { s (s-c) } } `
AREA OF A TRIANGLE:
` \ triangle = Area `
` \ triangle = Area `
3 SIDES GIVEN:
` \ triangle = \ sqrt { s (s-a) (s-b) (s-c) } `
` s = \ frac { a + b + c } 2 `
` \ triangle = \ sqrt { s (s-a) (s-b) (s-c) } `
` s = \ frac { a + b + c } 2 `
2 SIDES INCLUDED ANGLE GIVEN:
` \ triangle = \ frac { bc \ sin \ left( \ alpha \ right) } 2 `
` \ triangle = \ frac { ac \ sin \ left( \ beta \ right) } 2 `
` \ triangle = \ frac { ab \ sin \ left( \ gamma \ right) } 2 `
` \ triangle = \ frac { bc \ sin \ left( \ alpha \ right) } 2 `
` \ triangle = \ frac { ac \ sin \ left( \ beta \ right) } 2 `
` \ triangle = \ frac { ab \ sin \ left( \ gamma \ right) } 2 `
1 SIDES INCLUDED 2 ANGLES GIVEN:
` \
triangle = \ frac { a^2 \ sin \ left( \ beta \ right) \ sin \ left( \
gamma \ right) } { 2 \ sin \ left( \ alpha \ right) } `
` \
triangle = \ frac { b^2 \ sin \ left( \ alpha \ right) \ sin \ left( \
gamma \ right) } { 2 \ sin \ left( \ beta \ right) } `
` \
triangle = \ frac { c^2 \ sin \ left( \ alpha \ right) \ sin \ left( \
beta \ right) } { 2 \ sin \ left( \ gamma \ right) } `
` \
triangle = \ frac { a^2 \ sin \ left( \ beta \ right) \ sin \ left( \
gamma \ right) } { 2 \ sin \ left( \ alpha \ right) } `
` \
triangle = \ frac { b^2 \ sin \ left( \ alpha \ right) \ sin \ left( \
gamma \ right) } { 2 \ sin \ left( \ beta \ right) } `
` \
triangle = \ frac { c^2 \ sin \ left( \ alpha \ right) \ sin \ left( \
beta \ right) } { 2 \ sin \ left( \ gamma \ right) } `
CIRCUM-RADIUS R IN TERMS OF THE MEASURE OF A THREE SIDES OF A TRIANGLE:
`R \ = \ frac{ \ abc } { 4\triangle}`
`R \ = \ frac{ \ abc } { 4\triangle}`
THE IN-RADIUS OF A TRIANGLE:
` r = \ frac \triangle s`
` r = \ frac \triangle s`
