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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,559

Trigonometry Formulas

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

In a right-angled triangle,

Sinθ= Opposite Side/Hypotenuse

Cosθ= Adjacent Side/Hypotenuse

Tanθ= Sinθ/Cosθ = Opposite Side/Adjacent Side

Cosecθ = 1/Sinθ= Hypotenuse/Opposite Side

Secθ = 1/Cosθ = Hypotenuse/Adjacent Side

Cotθ = 1/tanθ = Cosθ/Sinθ = Adjacent Side/Opposite Side

SinθCosecθ = CosθSecθ = TanθCotθ = 1

Sin(90-θ) = Cosθ, Cos(90-θ) = Sinθ

Sin²θ + Cos²θ = 1

Tan²θ + 1 = Sec²θ

Cot²θ + 1 = Cosec²θ

Addition and subtraction formula:-

Sin(A+B) = SinACosB + CosASinB

Sin(A-B) = SinACosb - CosASinB

Cos(A+B) = CosACosB - SinASinB

Cos(A-B) = CosACosB + SinASinB

Tan(A+B) = (TanA+TanB)/(1-TanATanB)

Tan(A-B) = (TanA - TanB)/(1+TanATanB)

Cot (A+B) = (CotACotB-1)/(CotA + CotB)

Cot(A-B) = (CotACotB+1)/(CotB-CotA)

Sin(A+B)+Sin(A-B) = 2SinACosB

Sin(A+B)-Sin(A-B) = 2CosASinB

Cos(A+B)+Cos(A-B) = 2CosACosB

Cos(A-B) - Cos(A-B) = 2SinASinB

SinC + SinD = 2Sin[(C+D)/2]Cos[(C-D)/2]

SinC - SinD = 2Cos[(C+D)/2]Sin[(C-D)/2]

CosC + CosD = 2Cos[(C+D)/2]Cos[(C-D)/2]

CosC - CosD = 2Sin[(C+D)/2]Sin[(D-C)/2]

Sin2θ = 2SinθCosθ = (2tanθ)/(1+tan²θ)

Cos2θ = Cos²θ - Sin²θ = 2Cos²θ - 1= 1 - 2Sin²θ =

(1-tan²θ)/(1+tan²θ)

Tan2θ = 2tan θ/(1-tan²θ)

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

(Angles are given in degrees, 90 degrees, 180 degrees etc.)

I.

Sin(-θ)=-Sinθ

Cos(-θ) = Cosθ

tan(-θ) = -tanθ

cot(-θ) = -cotθ

sec(-θ) = secθ

cosec(-θ)= - cosecθ

II.

sin(90-θ) = cosθ

cos(90-θ) = sinθ

tan(90-θ) = cotθ

cot(90-θ) = tanθ

sec(90-θ) = cosecθ

cosec(90-θ) = secθ

III.

sin(90+θ) = cosθ

cos(90+θ) = -sinθ

tan(90+θ) = -cotθ

cot(90+θ) = -tanθ

sec(90+θ) = -cosecθ

cosec(90+θ) = secθ

IV.

sin(180-θ) = sinθ

cos(180-θ) = -cosθ

tan(180-θ) = -tanθ

cot(180-θ) = cotθ

sec(180-θ) = -secθ

cosec(180-θ) = cosecθ

V.

sin(180+θ) = -sinθ

cos(180+θ) = -cosθ

tan(180+θ) = tanθ

cot(180+θ) = cotθ

sec(180+θ) = -secθ

cosec(180+θ) = -cosecθ

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

**Formulas which express the sum or difference in product**

**Formulae which express products as sums or difference of Sines and Cosines**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

**Trignometric ratios of Multiple Angles**

**Trignometric ratios of 3θ**

**Trignometric ratios of sub-multiple angles**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

**Properties of Inverse Trignometric Functions**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

**Properties of Triangles**

**Sine Formula (or Law of Sines)**

In any ΔABC,

**Cosine Formula (or Law of Cosines)**

In any ΔABC,

These formulas are also written as

**Projection formulas**

In any ΔABC,

**Half-Angles and Sides**

In any ΔABC,

**Area of a Triangle**

Hero's fromula

**Incircle and Circumcircle**

A circle which touches the three sides of a traingle internally is called the incircle.The center of the circle is called the incentre and the raidus is called the inradius.

If r is the inradius, then

The circle which passes through the vertices of a triangle is called the circumcircle of a triangle or circumscribing circle. The centre of this circle is the circumcentre and the radius of the circumcircle is the circumradius.

If R is the circumradius, then

If Δ is the area of the triangle,

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

**Hyperbolic Functions**

**Relation between circular and hyperbolic functions**

**Addition formulas for Hyperbolic functions**

**Periods of hyperbolic functions**

**Inverse Hyperbolic functions**

Character is who you are when no one is looking.

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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

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**Devantè****Real Member**- Registered: 2006-07-14
- Posts: 6,400

*Last edited by Devanté (2006-10-10 07:58:28)*

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**kavish2000****Member**- Registered: 2007-09-07
- Posts: 1

ganesh wrote:

Formulas which express the sum or difference in product

Hey ganesh

i just joined this forum. i am an engineer. Am preparing for CAT exam.

am sure u knw abt CAT. (its this november) . So i was looking for some really interesting geometry and number system stuff

like some patterns or some formulaes

etc

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,065

Hi kavish2000,

I am sorry for the delay in replying,

Yes, I know about CAT, the general questioning pattern etc.

But what exactly do you want to know in Geometry and Number stuff?

Any prearatory CAT book gives the basics. And CAT doesn't require the highest level of Geometry skills or Number theory skills. Beig familiar with UG level mathematics and to some extent PG level would do.

The wikipedia always has much interesting stuff in geometric and number system, provided you know what exactly are the search words you use, and depending on your luck when choosing the relevance percentage.

There are some other interesting forums, sites on the net. If I were you, I would exhaust all search engines, and just hope I am lucky!

Not many engineers pursue the CAT, and most of them who do are likely to be successful. My cousin is one, just about my age, and he's now with an MNC at middle/top management.

My good wishes to you for the CAT, its November already.

Character is who you are when no one is looking.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,585

New Formula:

tan(2u)=2/(cot(u)-tan(u))

**igloo** **myrtilles** **fourmis**

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

*Last edited by Daniel123 (2009-05-10 05:47:10)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,914

Hi;

At the request of a member I have cleaned this thread to only reflect proven formulas. Some errors as pointed out by John E. Franklin have now been checked and corrected.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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