List of Trigonometric Formulae

In this article, we are going to list down all the important Trigonometric Formulas. Try to remember these.

For the purpose of objective-type aptitude examinations, we need not know how to drive them. But we should remember them, and develop our capability to use the right formula when required.

Relation among Trigonometric identities

When angles are same

Type 1

sin θ × cosec θ = 1
cos θ × sec θ = 1
tan θ × cot θ = 1

Type 2

$si{n}^2$ θ + $co{s}^2$ θ = 1
$se{c}^2$ θ - $ta{n}^2$ θ = 1
$cose{c}^2$ θ - $co{t}^2$ θ = 1

Type 3

If sin θ + cosec θ = 2, then:
$si{n}^n\theta +cose{c}^n\theta$ = 2 (where n is a natural number)

If cos θ + sec θ = 2, then:
$co{s}^n\theta +se{c}^n\theta$ = 2 (where n is a natural number)

If tan θ + cot θ = 2, then:
$ta{n}^n\theta +co{t}^n\theta$ = 2 (where n is a natural number)

Type 4

If:
I. a sin θ + b cos θ = c and
II. b sin θ – a cos θ = d or a cos θ - b sin θ = d
Then, $c^2+d^2$ = $a^2+b^2$

If:
I. sin θ + cos θ = c and
II. sin θ – cos θ = d
Then, $c^2+d^2$ = 2

If:
I. a sec θ + b tan θ = c; b sec θ + a tan θ = d, or
II. a sec θ - b tan θ = c; b sec θ - a tan θ = d
Then, $a^2-b^2$ = $c^2-d^2$

If:
I. a cosec θ + b cot θ = c; b cosec θ + a cot θ = d, or
II. a cosec θ - b cot θ = c; b cosec θ - a cot θ = d
Then, $a^2-b^2$ = $c^2-d^2$

When sum of angles is 90°

If θ + ɸ = 90°, then:

sin θ × sec ɸ = 1
or, sin θ = cos ɸ

cos θ × cosec ɸ = 1
or, cos θ = sin ɸ

tan θ × tan ɸ = 1
or, tan θ = cot ɸ

cot θ × cot ɸ = 1
or, cot θ = tan ɸ

If θ + ɸ + α = 90°, then:

(tan θ × tan ɸ) + (tan ɸ × tan α) + (tan α × tan θ) = 1

cot θ + cot ɸ + cot α = cot θ × cot ɸ × cot α

When sum of angles is 180°

If θ + ɸ = 180°, then:

sin θ × cosec ɸ = 1

If θ + ɸ + α = 180° (i.e. we are talking about a triangle), then:

tan θ + tan ɸ + tan α = tan θ × tan ɸ × tan α

(cot θ × cot ɸ) + (cot ɸ × cot α) + (cot θ × cot α) = 1

When sum of angles is 45° or 225°

If θ + ɸ = 45° or 225°, then:

(1 + tan θ) (1 + tan ɸ) = 2

(cot θ - 1) (cot ɸ - 1) = 2, Or
(1 - cot θ) (1 - cot ɸ) = 2

When difference of angles is 45° or 225°

If θ - ɸ = 45° or 225°, then:

(1 + tan θ) (1 - tan ɸ) = 2

(1 - cot θ) (1 + cot ɸ) = 2

Sum and Difference formulae

Type 1

sin (A ± B) = sin A . cos B ± cos A . sin B

cos (A ± B) = cos A . cos B ∓ sin A . sin B

tan (A ± B) = $\frac{tanA\pm tanB}{1\mp tanA.tanB}$

cot (A ± B) = $\frac{cotA.cotB\mp 1}{cotA\pm cotB}$

Type 2

sin (A + B) + sin (A - B) = 2 sin A . cos B
sin (A + B) - sin (A - B) = 2 cos A . sin B
cos (A + B) + cos (A - B) = 2 cos A . cos B
cos (A - B) - cos (A + B) = 2 sin A . sin B

Type 3

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

Type 4

sin A + sin B = 2 sin [$\frac{A+B}{2}$] . cos [$\frac{A-B}{2}$]

sin A – sin B = 2 cos [$\frac{A+B}{2}$] . sin [$\frac{A-B}{2}$]

cos A + cos B = 2 cos [$\frac{A+B}{2}$] . cos [$\frac{A-B}{2}$]

cos A – cos B = 2 sin [$\frac{A+B}{2}$] . sin [$\frac{B-A}{2}$]

Trigonometric ratios of Angle Multiples

sin

sin (2θ) = 2 sin θ cos θ = $\frac{2tan\theta }{1+ta{n}^2\theta }$

sin (3θ) = $3sin\theta -4si{n}^3\theta =sin\theta (-1+4co{s}^2\theta )$

sin (4θ) = $cos\theta (4sin\theta -8si{n}^3\theta )=sin\theta (-4cos\theta +8co{s}^3\theta )$

sin (5θ) = $5sin\theta -20si{n}^3\theta +16si{n}^5\theta =sin\theta (1-12co{s}^2\theta +16co{s}^4\theta )$

cos

cos (2θ) = $co{s}^2\theta -si{n}^2\theta =2co{s}^2\theta -1=1-2sin²\theta =\frac{1-tan²\theta }{1+tan²\theta }$

cos (3θ) = $co{s}^3\theta -3cos\theta si{n}^2\theta =4co{s}^3\theta -3cos\theta$

cos (4θ) = $co{s}^4\theta -6co{s}^2\theta si{n}^2\theta +si{n}^4\theta =1-8co{s}^2\theta +8co{s}^4\theta$

cos (5θ) = $co{s}^5\theta -10co{s}^3\theta si{n}^2\theta +5cos\theta si{n}^4\theta =5cos\theta -20co{s}^3\theta +16co{s}^5\theta$

tan

tan (2θ) = $\frac{2tan\theta }{1-ta{n}^2\theta }$

tan (3θ) = $\frac{3tan\theta -ta{n}^3\theta }{1-3ta{n}^2\theta }$

tan (4θ) = $\frac{4tan\theta -4ta{n}^3\theta }{1-6ta{n}^2\theta +ta{n}^4\theta }$

Morri’s law

sin θ . sin(60° - θ) . sin (60° + θ) = $\frac{1}{4}$ sin 3θ

cos θ . cos(60° - θ) . cos (60° + θ) = $\frac{1}{4}$ cos 3θ

tan θ . tan (60° - θ) . tan (60° + θ) = tan 3θ

cot θ . cot (60° - θ) . cot (60° + θ) = cot 3θ

Maximum/Minimum Values

• Maximum value of $si{n}^n\theta co{s}^n\theta =(1/2{)}^n$

• Minimum value of $si{n}^n\theta co{s}^n\theta =-(1/2{)}^n$ or 0 (if n is even)

• Maximum value of $si{n}^n\theta +co{s}^n\theta =1$ (always)

• Minimum value of $si{n}^n\theta +co{s}^n\theta$ will be when θ = 45°

• Maximum value of a sin θ ± b cos θ = $\surd (a^2+b^2)$
  Minimum value of a sin θ ± b cos θ = – $\surd (a^2+b^2)$

• Maximum value of $asi{n}^2\theta +bco{s}^2\theta$ = a (If a>b) or b (If b>a)

• Minimum value of $asi{n}^2\theta +bco{s}^2\theta$ = b (If a>b) or a (If b>a)

• Minimum value of $asi{n}^2\theta +bcose{c}^2\theta =2\surd ab$ when b ≤ a, Or a + b, when b ≥ a

• Minimum value of $aco{s}^2\theta +bse{c}^2\theta =2\surd ab$ when b ≤ a, Or a + b, when b ≥ a

• Minimum value of $ata{n}^2\theta +bco{t}^2\theta =2\surd ab$

• Minimum value of $ase{c}^2\theta +bcose{c}^2\theta =(\surd a+\surd b{)}^2$