TRIGONOMETRY: ALL IMPORTANT FORMULAS

BASIC PRINCIPLE:-

In the given diagram, a right-angled triangle ABC is shown. Let's consider angle ACB as `theta`, then its opposite side AB will become the perpendicular(P), side BC will become the base(B), and side AC will become the hypotenuse(H). Then the respective ratios of trigonometry will be,

`sin` `theta` = `frac{P}{H}`

`cos` `theta` = `frac{B}{H}`

`tan` `theta` = `frac{P}{B}`

TRIGONOMETRIC IDENTITIES:-

`sin^{2}``theta` + `cos^{2}``theta` = 1

`sec^{2}``theta` = 1 + `tan^{2}``theta`

`cosec^{2}``theta` = 1 + `cot^{2}``theta`

SIGN CONVENTION:-

`sin` (-`theta`) = -`sin` `theta`

`cos` (-`theta`) = `cos` `theta`

`tan` (-`theta`) = -`tan` `theta`

QUADRANT SYSTEM AND CONVERSION RULES:-

In the given diagram, the four quadrants for the trigonometric functions have been shown. The two axes make a partition between the quadrants on the basis of ranges. The sign convention of ratios changes from one quadrant to another and in few cases they also get transformed from one to another.

In quadrant Ⅰ, all the ratios have positive signs.

In quadrant Ⅱ, only the sine ratio and its opposite, the cosecant ratio, are positive and there is also a ratio transformation rule, which is as follows

`sin` (90° + `theta`) = `cos` `theta`

In quadrant Ⅲ, only the tangent ratio and its opposite, the cotangent ratio, are positive and there is also a ratio transformation rule, which is as follows

`sin` (180° + `theta`) = -`sin` `theta`

In quadrant Ⅳ, only the cosine ratio and its opposite, the secant ratio, are positive and there is also a ratio transformation rule, which is as follows

`sin` (270° + `theta`) = -`cos` `theta`

ADDITION & SUBTRACTION OF ANGLES:-

`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) = `frac{tanA + tanB}{1 - tanAtanB}`

`tan`(A - B) = `frac{tanA - tanB}{1 + tanAtanB}`

DOUBLE & TRIPLE OF ANGLES:-

`sin`2A = 2`sin`A`cos`A

`cos`2A = `cos^{2}`A - `sin^{2}`A = 2`cos^{2}`A - 1 = 1 - 2`sin^{2}`A

`tan`2A = `\frac{2tanA}{1- tan^{2}A}`

`sin`3A = 3`sin`A - 4`sin^{3}`A

`cos`3A = 4`cos^{3}`A - 3`cos`A

`tan`3A = `\frac{3tanA - tan^{3}A}{1 - 3tan^{2}A}`

ADDITION & SUBTRACTION OF RATIOS:-

`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{A-B}{2}`)

INVERSE TRIGONOMETRIC FUNCTIONS:-

`sin^{-1}`x = `cosec^{-1}`(`\frac{1}{x}`)

`cos^{-1}`x = `sec^{-1}`(`\frac{1}{x}`)

`tan^{-1}`x = `cot^{-1}`(`\frac{1}{x}`), for x ＞ 0

`tan^{-1}`x = -π + `cot^{-1}`(`\frac{1}{x}`), for x < 0

`sin^{-1}`(-x) = -`sin^{-1}`x

`cos^{-1}`(-x) = π - `cos^{-1}`x

`tan^{-1}`(-x) = -`tan^{-1}`x

`cosec^{-1}`(-x) = -`cosec^{-1}`x

`sec^{-1}`(-x) = π - `sec^{-1}`x

`cot^{-1}`(-x) = π -`cot^{-1}`x

`sin^{-1}`x + `cos^{-1}`x = `\frac{π}{2}`

`tan^{-1}`x + `cot^{-1}`x = `\frac{π}{2}`

`cosec^{-1}`x + `sec^{-1}`x = `\frac{π}{2}`

`sin^{-1}`x + `sin^{-1}`y = `sin^{-1}`(x`\sqrt{1-y^{2}}` + y`\sqrt{1-x^{2}}`)

`sin^{-1}`x - `sin^{-1}`y = `sin^{-1}`(x`\sqrt{1-y^{2}}` + y`\sqrt{1-x^{2}}`)

`cos^{-1}`x + `cos^{-1}`y = `cos^{-1}`(x`\sqrt{1-y^{2}}` + y`\sqrt{1-x^{2}}`)

`cos^{-1}`x - `cos^{-1}`y = `cos^{-1}`(x`\sqrt{1-y^{2}}` + y`\sqrt{1-x^{2}}`)

`tan^{-1}`x + `tan^{-1}`y = `\tan^{-1}(\frac{x + y}{1 - xy})`

`tan^{-1}`x - `tan^{-1}`y = `\tan^{-1}(\frac{x - y}{1 + xy})`