Trigonometry Formulaes

8/19/2019 Trigonometry Formulaes

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. TrigonometryHello Friends there is no limit of Mathematics. For being a masterpiece a lot of practice is required.

Solve as many problems as you can.

This Note contains the basic formulas followed by method of finding maximum and minimum.AFTER LOT OF PRACTICE YOU REACH A STAGE THAT WITHOUT SEEING ANY FORMULA YOU WILL DIRECTLY SOLVE

THE QUESTION.

Answer by seeing the option or eliminating the wrong option is very important in Comptt.Exam (Speciallly in

Trigonometry and Algebra.)

Most of us face problem in determining the Maximum and Minimum value of given trigonometric expression.

I have given the different approach [CALCULAS] (the guys who have Mathematics as a part of their 12th standard)

are expected to know this.If forgotten the rules then pls refer last part of this note

But others who don’t have any idea about Calculus, I request them to learn the basic identities of Differentiation

anyhow.

It will take 2-3 days.

That would be very beneficial for Trigonometry/Algebra & will save a lot of time.

For SSC Pattern Trigonometry Questions you may follow any class 10th standard book which provide you lot of

questions

Exampundit is always here to help you and provide you practice exercise too.

Basic Ratios

Example: Write expressions for the sine, cosine, and tangent of A.

The length of the leg opposite A is a. The length of the leg adjacent to A is b, and the length of the hypotenuse is c.

The sine of the angle is g iven by the ratio "opposite over hypotenuse". So,


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The cosine is given by the ratio "adjacent over hypotenuse".

The tangent is given by the ratio "opposite over adjacent".

Other Trigonometric Ratios

The other common trigonometric ratios are:

Pythagorean Identities

Co-Function Identities

Even-Odd Identities


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Sum-Difference Formulas

Double/Triple Angle Formulas

Sin 3A = 3 Sin A - 4 sin³A

Cos 3A = 4 Cos³A - 3 Cos A

tan 3A = (3 tan A - tan³ A)/(1-3tan²A)

Power-Reducing/Half Angle Formulas


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Sum-to-Product Formulas

Product-to-Sum Formulas


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Values at Different Angles

ASTC (All Student Take Coffee) Rule


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At 90 and 270 degrees Sin--- Cos

Sec---Cosec

Tan---Cot

& Vice-Versa.But sign changes according to above rule

At 180 and 360 degrees no change is made

But sign changes as per rule.

Example:

Tan(180-x)=?

..

As we know that at 180 Tan remains Tan so ans will be Tan X for sure

But sign will be –ve because 180-x lies in second quardrant where only sin is +ve

So Final Answer will be –Tan X

..

Cos(270-x)=?

As we know at 270 cos changes to Sin so ans will be sin x

But sign will be –ve because 270-x lies in 3rd quardrant where only Tan is +ve

So final ans –Sin x


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Radian to Degree

1 rad = 180º/π

Degree to Radian

1º = π/180º

Maximum and Minimum Values

Min Max

sin θ, sin 2θ, sin 9θ …. sin nθ

-1 +1

-1 ≤ Sin nθ ≤ 1

cos θ, cos 4θ , cos 7θ … cos nθ -1 ≤ Cos nθ ≤ 1

sin2 θ , sin2 4θ , sin2 9θ …sin2 nθ

0 +1

Can be written as0 ≤ Sin2 nθ ≤ 1

cos2 θ , cos2 3θ , cos2 8θ … cos2 nθ 0 ≤ Cos2 nθ ≤ 1

Sin θ Cos θ -1/2 +1/2 -1/2 ≤ Sin θ Cos θ ≤ ½

Other Varients

1.

a sin θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }

2.

a sin θ ± b sin θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }

3.

a cos θ ± b cos θ = ±√ (a2 + b2 ) { for min. use – , for max. use + }

4.

Min. value of (sin θ cos θ)n = (½)


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In case of sec2x, cosec2x, cot2x and tan2x, we cannot find the maximum value because they can have infinity

as their maximum value. So in question containing these trigonometric identities, you will be asked to find the

minimum values only. The typical question forms are listed below:

In this type, we will give you the explanation of question which are different from type-1, type-2 and type-3. If you find this kind

of questions, you will have to convert these questions into type-1,2 or 3 by using trigonometric formulas

Example - Find the Minimum Value of Sec 2x + cosec 2x

Sol - 1 + tan 2x + cosec 2x --------------------------------------------(Sec2x = 1 + tan 2x)

= 1+ tan2

x + 1 + cot2

x ------------------------------------------------(cosec2

x = 1 + cot2

x )=2 + tan 2x + cot 2x---------------------------------------------------apply type-3 formula

=2 + 2 √ 1 x 1

= 2 + 2

=4

The best method to find Maximum and Minimum Values is Calculus

Approach

1-Find the derivative(d/dx) of the given expression

2-Put that differentiated value=0

3-Find the value of (x, theta or whatever) that satisfies it

4-Put the value in the given expression and find maximum and minimum

Example (1) Maximum and Minimum value of sin x +cos x

Solution- Step 1- Take derivative

d/dx(sin x + cos x)

=cos x –sin x

Step 2-Put this equal to 0


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Cos x – sin x =0

So cos x = sin x

As we clearly know that x=45 degree satisfies this

Put x=45 degrees (+ve for maximum value)

Sin 45+ cos 45 = 1/under root 2 + 1/ under root 2=root 2=Maximum value

&

Minimum Value =- root 2

Graphical approach

Arjit

Trigonometry Formulaes

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