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

$\ cot^2 \ left( \theta\right) \ = \ \ csc^2 \ left( \theta\right) \ - \ 1$

=====

NEGATIVE(-ve) ANGLE FORMULAS:

$\ sin \ left( -\theta\right) \ =\ - \ sin\ left( \theta\right)$

$\ cos \ left( -\theta\right) \ = \ \ cos\ left( \theta\right)$

$\ tan \ left( -\theta\right) \ = \ - \ tan\ left( \theta\right)$

$\ cot \ left( -\theta\right) \ = \ - \ cot\ left( \theta\right)$

$\ sec \ left( -\theta\right) \ =\ \ sec\ left( \theta\right)$

$\ csc \ left( -\theta\right) \ = \ - \ csc\ left( \theta\right)$

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

$\ cos \ left( 2\theta\right) \ = \ \ frac {1\ - \ \ tan^2\ left( \theta\right)}{1\ + \ \ tan^2\ left( \theta\right)}$

= = = = =

$\ tan \ left( 2\theta\right) \ = \ \ frac { 2 \ tan \ left( \theta\right)}{ 1-\ tan^2\ left( \theta\right)}$

HALF ANGLE FORMULAS:

$\ sin \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1- \ cos\ left( \ alpha \ right)} 2}$

$\ cos \ left( \ frac \alpha2\ right) \ = \ \ pm \ sqrt{ \ frac { 1+ \ cos\ left( \ alpha \ right)} 2}$

$\ tan \ left( \ frac \alpha2\ right) \ = \ pm \ sqrt{ \ frac{1 - \ cos\ left( \ alpha \ right)} {1 + \ cos\ left( \ alpha \ right)}}$

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:

$\ sin \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \ frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) + \ sin \ left( \ alpha - \ beta \ right) \ right]$

$\ cos \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = \ frac1\2 \ left[ \ sin \ left( \ alpha + \ beta \ right) - \ sin \ left( \ alpha - \ beta \ right) \ right]$

$\ cos \ left( \ alpha \ right) \ \ cos \ left( \ beta \ right) = \ frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) + \ cos \ left( \ alpha - \ beta \ right) \ right]$

$\ sin \ left( \ alpha \ right) \ \ sin \ left( \ beta \ right) = - \ frac1\2 \ left[ \ cos \ left( \ alpha + \ beta \ right) - \ cos \ left( \ alpha - \ beta \ right) \ right]$

SUM TO PRODUCT FORMULAS:

$\ sin \ left( \ upsilon \ right) + \ sin \ left( \ nu \ right) = 2 \ sin \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right)$

$\ sin \ left( \ upsilon \ right) -\ sin \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2 \ right)$

$\ cos \ left( \ upsilon \ right) + \ cos \ left( \ nu \ right) = 2 \ cos \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ cos \ left( \ frac { \ upsilon - \ nu} 2 \ right)$

$\ cos \ left( \ upsilon \ right) -\ cos \ left( \ nu \ right) = - 2 \ sin \ left( \ frac { \ upsilon + \ nu} 2 \ right) \ sin \ left( \ frac { \ upsilon - \ nu} 2 \ right)$

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

$a \ : \ b \ : \ c \ = \ \ sin\ left( \ alpha \ right) \ : \ \ sin \ left( \ beta \ right) \ : \ \ sin \ left( \ gamma \ right)$

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

$c^2 = a^2 + b^2 -2ab \ cos \ left( \ gamma \ right)$

HALF ANGLE FORMULAS:

$\ sin \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { bc } }$

$\ sin \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { ac } }$

$\ sin \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { ab } }$

$\ cos \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { s (s-a) } { bc } }$

$\ cos \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { s (s-b) } { ac } }$

$\ cos \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { s (s-c) } { ab } }$

$\ tan \ left( \ frac \alpha2\ right) \ = \ sqrt { \ frac { (s-b) \ (s-c) } { s (s-a) } }$

$\ tan \ left( \ frac \beta2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-c) } { s (s-b) } }$

$\ tan \ left( \ frac \gamma2\ right) \ = \ sqrt { \ frac { (s-a) \ (s-b) } { s (s-c) } }$

AREA OF A TRIANGLE:

$\ triangle = Area$

3 SIDES GIVEN:

$\ triangle = \ sqrt { s (s-a) (s-b) (s-c) }$

$s = \ frac { a + b + c } 2$

2 SIDES INCLUDED ANGLE GIVEN:

$\ triangle = \ frac { bc \ sin \ left( \ alpha \ right) } 2$

$\ triangle = \ frac { ac \ sin \ left( \ beta \ right) } 2$

$\ triangle = \ frac { ab \ sin \ left( \ gamma \ right) } 2$

1 SIDES INCLUDED 2 ANGLES GIVEN:

$\ triangle = \ frac { a^2 \ sin \ left( \ beta \ right) \ sin \ left( \ gamma \ right) } { 2 \ sin \ left( \ alpha \ right) }$

$\ triangle = \ frac { b^2 \ sin \ left( \ alpha \ right) \ sin \ left( \ gamma \ right) } { 2 \ sin \ left( \ beta \ right) }$

$\ triangle = \ frac { c^2 \ sin \ left( \ alpha \ right) \ sin \ left( \ beta \ right) } { 2 \ sin \ left( \ gamma \ right) }$

CIRCUM-RADIUS R IN TERMS OF THE MEASURE OF A THREE SIDES OF A TRIANGLE:

$R \ = \ frac{ \ abc } { 4\triangle}$

THE IN-RADIUS OF A TRIANGLE:

$r = \ frac \triangle s$

RADII OF E-CIRCLES OF A TRIANGLE:

$r_1 = \ frac \triangle {s-a}$

$r_2 = \ frac \triangle {s-b}$

$r_3 = \ frac \triangle {s-c}$

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Trigonometric Functions Formulas

Trigonometric Formulas:

Basic Formulas:

NEGATIVE(-ve) ANGLE FORMULAS:

DOUBLE ANGLE FORMULAS:

HALF ANGLE FORMULAS:

FUNDAMENTAL LAWS:

SUM TO PRODUCT FORMULAS:

LAW OF SINE:

LAW OF COSINE:

HALF ANGLE FORMULAS:

AREA OF A TRIANGLE:

$\ triangle = Area$

3 SIDES GIVEN:

$\ triangle = \ sqrt { s (s-a) (s-b) (s-c) }$

$s = \ frac { a + b + c } 2$

2 SIDES INCLUDED ANGLE GIVEN:

$\ triangle = \ frac { bc \ sin \ left( \ alpha \ right) } 2$

$\ triangle = \ frac { ac \ sin \ left( \ beta \ right) } 2$

$\ triangle = \ frac { ab \ sin \ left( \ gamma \ right) } 2$

1 SIDES INCLUDED 2 ANGLES GIVEN:

CIRCUM-RADIUS R IN TERMS OF THE MEASURE OF A THREE SIDES OF A TRIANGLE:

$R \ = \ frac{ \ abc } { 4\triangle}$

THE IN-RADIUS OF A TRIANGLE:

$r = \ frac \triangle s$

RADII OF E-CIRCLES OF A TRIANGLE:

$r_1 = \ frac \triangle {s-a}$

$r_2 = \ frac \triangle {s-b}$

$r_3 = \ frac \triangle {s-c}$

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Trigonometric Functions Formulas

Trigonometric Formulas:

Basic Formulas:

NEGATIVE(-ve) ANGLE FORMULAS:

DOUBLE ANGLE FORMULAS:

HALF ANGLE FORMULAS:

FUNDAMENTAL LAWS:

SUM TO PRODUCT FORMULAS:

LAW OF SINE:

LAW OF COSINE:

HALF ANGLE FORMULAS:

AREA OF A TRIANGLE:

\ triangle = Area \ triangle = Area

3 SIDES GIVEN:

\ triangle = \ sqrt { s (s-a) (s-b) (s-c) } \ triangle = \ sqrt { s (s-a) (s-b) (s-c) } s = \ frac { a + b + c } 2 s = \ frac { a + b + c } 2

2 SIDES INCLUDED ANGLE GIVEN:

1 SIDES INCLUDED 2 ANGLES GIVEN:

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

RADII OF E-CIRCLES OF A TRIANGLE:

r_1 = \ frac \triangle {s-a} r_1 = \ frac \triangle {s-a} r_2 = \ frac \triangle {s-b} r_2 = \ frac \triangle {s-b} r_3 = \ frac \triangle {s-c} r_3 = \ frac \triangle {s-c}

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$\ triangle = Area$

$\ triangle = \ sqrt { s (s-a) (s-b) (s-c) }$

$s = \ frac { a + b + c } 2$

$R \ = \ frac{ \ abc } { 4\triangle}$

$r = \ frac \triangle s$

$r_1 = \ frac \triangle {s-a}$

$r_2 = \ frac \triangle {s-b}$

$r_3 = \ frac \triangle {s-c}$