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Maths >> Trigonometry >> General Trigonometry Board Exam

if x=2Cosθ-Cos2θ and y=2Sinθ-Sin2θ prove that dy/dx=tan(3θ/2)

Asked By: SARABJEET
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Joshi sir comment

calculate dx/dθ and dy/dθ

then dy/dx = [2cosθ-2cos2θ]/[-2sinθ+2sin2θ]

now solve

use cosC-cosD = 2sin[(C+D)/2]sin[(D-C)/2]

and sinC+sinD = 2sin[(C+D)/2]cos[(C-D)/2]

Maths >> Trigonometry >> Trigonometrical Equations Olympiad

sinA + sinB + sinC = 4cosA/2.cosB/2.cosC/2

Asked By: AVI GARG
is this question helpfull: 3 0 submit your answer
Joshi sir comment

sinA + sinB + sinC = 2sin[(A+B)/2]cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2sin[(π-C)/2]cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2cosC/2cos[(A-B)/2] + 2sinC/2cosC/2

                                  = 2cosC/2* [cos[(A-B)/2]+[cos(A+B)/2]                     here sinC/2 = cos(A+B)/2

now use cosC + cosD = 2 cos[(C+D)/2]cos[(C-D)/2]

You should remember that the given formula is based on a triangle. 

Maths >> Calculus >> Limit Continuity and Differentiability IIT JEE

lim n--> ∞  4^n/n!

Asked By: HIMANSHU MITTAL
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Joshi sir comment

limn->∞ 4n/ n!

= limn->∞ [4/1][4/2][4/3][4/4][4/5][4/6][4/7]................................[4/n]

besides first four bracket, all brackets are real numbers less than 1 so their product will be zero for n tends to infinite

 

Maths >> Calculus >> Differentiation IIT JEE
Sir,is the function (2x+5) is differentiable everywhere in its domain set. If yes the what is the case with l2x+5l.
Asked By: GAURAV MAHATE
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Joshi sir comment

since first one is a straight line so at every point only one tangent is possible so it is differentiable

but second one which is modulus has a sharp turn at x = -5/2 so two tangents at x = -5/2 are possible so is not differentiable.

Maths >> Calculus >> Applications of differentiation IIT JEE

A window frame is shaped like a rectangle with an arch forming the top ( ie; a box with a semicircle on the top)

The vertical straight side is Y, the width at the base and at the widest part of the arch (Diam) is X.

I know that the perimeter is 4.

Find X if the area of the frame is at a maximum?

Asked By: ROBERT DEPANGHER
is this question helpfull: 3 2 read solutions ( 1 ) | submit your answer
Joshi sir comment

perimeter = 4 so 2Y + X + πX/2 = 4

and area = XY + [π(X/2)2]/2

put Y from 1st equation to 2nd we get A = X[4-X{1+(π/2)}]/2 + πX2/8

now calculate dA/dX and then compare it to zero

X will be 8/(4+π)

Maths >> Co ordinate Geometry >> General Coordinate Geometry Olympiad

 

For which integers n ≥ 3 does there exist a regular n-gon in the plane such that
all of its vertices have integer coordinates?
 
Asked By: BUDDHI PRAKASH
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Joshi sir comment

for n = 4, we can form a square with integral coordinates of all the vertices.

Maths >> Co ordinate Geometry >> General Coordinate Geometry Olympiad

 

Prove that the four projections of vertex A of the triangle ABC onto the exterior
and interior angle bisectors of  B and  C are collinear.
Asked By: BUDDHI PRAKASH
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Joshi sir comment

in the given diagram red lines are angle bisectors (interior and exterior)and X, Y, Z, W are the projections of A on these red lines

by angle bisector property all these 4 points will lie on the line BC (either externally or internally) so these points will be in a line

Maths >> Trigonometry >> Inverse Trigonometry IIT JEE

If cos-1x - cos-1(y/2) = α , then 4x2-4xycosα + y2 is equal to ?

Asked By: AMIT DAS
is this question helpfull: 12 6 submit your answer
Joshi sir comment

according to the given condition 

cos-1x  =  cos-1(y/2)  +  a

so x = cos ( cos-1y/2   +  a)

so x = y/2 cosa  -  sin cos-1y/2 + sina

so x = y/2 cosa  - [1-(y2/4)]1/2 + sina

so x - sina - y/2 cosa = -[1-(y2/4)]1/2

now square both side and solve

 

Maths >> Trigonometry >> Inverse Trigonometry IIT JEE

If  sin-1a + sin-1b+ sin-1c = π , then the value of { a√(1-a2) + b√(1-b2) + c√(1-c2)}

Asked By: AMIT DAS
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Joshi sir comment

let sin-1a = A ,    sin-1b = B    and    sin-1c= C 

so  { a√(1-a2) + b√(1-b2) + c√(1-c2)}   = sinAcosA + sinBcosB + sinCcosC   = 1/2 (sin2A+sin2B+sin2C)

                                                                                                          = 1/2 (4sinAsinBsinC)

                                                                                                          = 2abc                                                    

Maths >> Trigonometry >> Inverse Trigonometry IIT JEE

If    [cot‾1x] + [cos‾1x] = 0 , then complete set of value of x is ( [ * ] is GIF)?

Asked By: AMIT DAS
is this question helpfull: 7 4 submit your answer
Joshi sir comment

These are the graphs for cot-1x and cos-1x   

violet for cos and green for cot

from graph it is clear that cos part with integer function will be 0 for cos1<x≤1

and cot part with integer function will be 0 for cot1<x<∞

so answer will be cot1<x≤1

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