# 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

These formulae can’t be applied if the two angles are different (θ ≠ Φ).
For example, sin θ × cosec Φ ≠ 1

#### Type 2

$sin^2$ θ + $cos^2$ θ = 1
$sec^2$ θ - $tan^2$ θ = 1
$cosec^2$ θ - $cot^2$ θ = 1

These formulae can’t be applied if the two angles are different (θ ≠ Φ).
For example, $sin^2$ θ + $cos^2$ Φ ≠ 1

#### Type 3

If sin θ + cosec θ = 2, then:
$sin^n θ + cosec^n θ$ = 2 (where n is a natural number)

If cos θ + sec θ = 2, then:
$cos^n θ + sec^n θ$ = 2 (where n is a natural number)

If tan θ + cot θ = 2, then:
$tan^n θ + cot^n θ$ = 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) = $\frac{tan A \hspace{1ex} ± \hspace{1ex} tan B}{1 \hspace{1ex} ∓ \hspace{1ex} tan A \hspace{1ex} . \hspace{1ex} tan B}$

cot (A ± B) = $\frac{cot A \hspace{1ex} . \hspace{1ex} cotB \hspace{1ex} ∓ \hspace{1ex} 1}{cot A \hspace{1ex} ± \hspace{1ex} cot B}$

### 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 = $\frac{2 \hspace{1ex} tan θ}{1 + tan^2θ}$

sin (3θ) = $3 \hspace{1ex} sin θ - 4 \hspace{1ex} sin^3 θ = sin θ (- 1 + 4 \hspace{1ex} cos^2 θ)$

sin (4θ) = $cos θ (4 \hspace{1ex} sin θ - 8 \hspace{1ex} sin^3 θ) = sin θ (- 4 \hspace{1ex} cos θ + 8 \hspace{1ex} cos^3 θ)$

sin (5θ) = $5 \hspace{1ex} sin θ - 20 \hspace{1ex} sin^3 θ + 16 \hspace{1ex} sin^5 θ = sin θ (1 - 12 \hspace{1ex} cos^2 θ + 16 \hspace{1ex} cos^4 θ)$

### cos

cos (2θ) = $cos^2 θ - sin^2 θ = 2 \hspace{1ex} cos^2 θ - 1 = 1 - 2 \hspace{1ex} sin² θ = \frac{1 - tan² θ}{1 + tan² θ}$

cos (3θ) = $cos^3 θ - 3 \hspace{1ex} cos θ sin^2 θ = 4 \hspace{1ex} cos^3 θ - 3 \hspace{1ex} cos θ$

cos (4θ) = $cos^4 θ - 6 \hspace{1ex} cos^2 θ sin^2 θ + sin^4 θ = 1 - 8 \hspace{1ex} cos^2 θ + 8 \hspace{1ex} cos^4 θ$

cos (5θ) = $cos^5 θ - 10 \hspace{1ex} cos^3 θ sin^2 θ + 5 \hspace{1ex} cos θ sin^4 θ = 5 \hspace{1ex} cos θ - 20 \hspace{1ex} cos^3 θ + 16 \hspace{1ex} cos^5 θ$

### tan

tan (2θ) = $\frac{2 \hspace{1ex} tan θ}{1 - tan^2 θ}$

tan (3θ) = $\frac{3 \hspace{1ex} tan θ - tan^3 θ}{1 - 3 \hspace{1ex} tan^2 θ}$

tan (4θ) = $\frac{4 \hspace{1ex} tan θ - 4 \hspace{1ex} tan^3 θ}{1 - 6 \hspace{1ex} tan^2 θ + tan^4 θ}$

### 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

The value of sec & cosec can be anything between -∞ to ∞. However, it can’t be between -1 and 1 (thouggh it can be -1 and 1).

That is, the range of value of sec & cosec = ∞ - (-1, 1)

• Maximum value of $sin^n θ \hspace{1ex} cos^n θ = (1/2)^n$

• Minimum value of $sin^n θ \hspace{1ex} cos^n θ = -(1/2)^n$ or 0 (if n is even)

• Maximum value of $sin^n θ + cos^n θ = 1$ (always)

• Minimum value of $sin^n θ + cos^n θ$ will be when θ = 45°

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

• Maximum value of $a \hspace{1ex} sin^2 θ + b \hspace{1ex} cos^2 θ$ = a (If a>b) or b (If b>a)

• Minimum value of $a \hspace{1ex} sin^2 θ + b \hspace{1ex} cos^2 θ$ = b (If a>b) or a (If b>a)

• Minimum value of $a \hspace{1ex} sin^2 θ + b \hspace{1ex} cosec^2 θ = 2√ab$ when b ≤ a, Or a + b, when b ≥ a

• Minimum value of $a \hspace{1ex} cos^2 θ + b \hspace{1ex} sec^2 θ = 2√ab$ when b ≤ a, Or a + b, when b ≥ a

• Minimum value of $a \hspace{1ex} tan^2 θ + b \hspace{1ex} cot^2 θ = 2√ab$

• Minimum value of $a \hspace{1ex} sec^2 θ + b \hspace{1ex} cosec^2 θ = (√a + √b)^2$

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