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k0ka [10]
2 years ago
7

Find the value of cosAcos2Acos3A...........cos998Acos999A where A=2π/1999

Mathematics
2 answers:
Lady bird [3.3K]2 years ago
7 0
Hello,

Here is the demonstration in the book Person Guide to Mathematic by Khattar Dinesh.

Let's assume
P=cos(a)*cos(2a)*cos(3a)*....*cos(998a)*cos(999a)
Q=sin(a)*sin(2a)*sin(3a)*....*sin(998a)*sin(999a)

As sin x *cos x=sin (2x) /2

P*Q=1/2*sin(2a)*1/2sin(4a)*1/2*sin(6a)*....
         *1/2* sin(2*998a)*1/2*sin(2*999a) (there are 999 factors)
= 1/(2^999) * sin(2a)*sin(4a)*...
     *sin(998a)*sin(1000a)*sin(1002a)*....*sin(1996a)*sin(1998a)
 as sin(x)=-sin(2pi-x) and 2pi=1999a

sin(1000a)=-sin(2pi-1000a)=-sin(1999a-1000a)=-sin(999a)
sin(1002a)=-sin(2pi-1002a)=-sin(1999a-1002a)=-sin(997a)
...
sin(1996a)=-sin(2pi-1996a)=-sin(1999a-1996a)=-sin(3a)
sin(1998a)=-sin(2pi-1998a)=-sin(1999a-1998a)=-sin(a)

So  sin(2a)*sin(4a)*...
     *sin(998a)*sin(1000a)*sin(1002a)*....*sin(1996a)*sin(1998a)
= sin(a)*sin(2a)*sin(3a)*....*sin(998)*sin(999) since there are 500 sign "-".

Thus
P*Q=1/2^999*Q or Q!=0 then
P=1/(2^999)

       








wel2 years ago
3 0

Answer:

The value of given expression is \frac{1}{2^{999}}.

Step-by-step explanation:

The given expression is

\cos A\cos 2A\cos 3A...........\cos 998A\cos 999A

where, A=\frac{2\pi}{1999}

Let as assume,

P=\cos A\cos 2A\cos 3A...........\cos 998A\cos 999A

Q=\sin A\sin 2A\sin 3A...........\sin 998A\sin 999A

2^{999}PQ=2^{999}(\cos A\cos 2A.........\cos 999A)(\sin A\sin 2A........\sin 999A)

2^{999}PQ=(2\cos A\sin A)(2\cos 2A\sin 2A)...........(2\cos 998A\sin 998A)(2\cos 999A\sin 999A)

Using the formula, 2\sin x\cos x=\sin 2x, we get

2^{999}PQ=\sin 2A\sin 4A......\sin 1998A

2^{999}PQ=[\sin 2A\sin 4A......\sin 998A][-\sin(2\pi -1000A)][-\sin(2\pi -1002A)]...[-\sin(2\pi -1998A)]             .... (1)

Now,

-\sin(2\pi -1000A)=-\sin(2\pi -1000(\frac{2\pi}{1999}))

-\sin(2\pi -1000A)=-\sin(\frac{2\pi\cdot 999}{1999})

-\sin(2\pi -1000A)=-\sin 999A

So, equation (1) can be written as

2^{999}PQ=[\sin 2A\sin 4A......\sin 998A][\sin 999A\sin 997...\sin A]

2^{999}PQ=\sin A\sin 2A\sin 3A...........\sin 998A\sin 999A

2^{999}PQ=Q

Divide both sides by Q.

2^{999}P=1

Divide both sides by 2^{999}

P=\frac{1}{2^{999}}

Therefore the value of given expression is \frac{1}{2^{999}}.

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zysi [14]

Known facts

  • w = 6
  • p = -5

8w - 2p = 8(6) - 2(-5) = 48 + 10 = 58

Answer: 58

Hope that helps!

4 0
2 years ago
Read 2 more answers
<img src="https://tex.z-dn.net/?f=%5Csqrt%7B%28-81%29x%5E%7B2%7D%20%7D" id="TexFormula1" title="\sqrt{(-81)x^{2} }" alt="\sqrt{(
SSSSS [86.1K]

Answer:

We conclude that:

\sqrt{\left(-81\right)x^2}=9ix

Step-by-step explanation:

Given the radical expression

\sqrt{\left(-81\right)x^2}

simplifying the expression

\sqrt{\left(-81\right)x^2}

Remove parentheses:  (-a) = -a

\sqrt{\left(-81\right)x^2}=\sqrt{-81x^2}

Apply radical rule:   \sqrt{-a}=\sqrt{-1}\sqrt{a},\:\quad \mathrm{\:assuming\:}a\ge 0

                 =\sqrt{-1}\sqrt{81x^2}

Apply imaginary number rule:  \sqrt{-1}=i

                 =i\sqrt{81x^2}

Apply radical rule:   \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0

                  =\sqrt{81}i\sqrt{x^2}

                  =9i\sqrt{x^2}

Apply radical rule:  \sqrt[n]{a^n}=a,\:\quad \mathrm{\:assuming\:}a\ge 0

                  =9ix

Therefore, we conclude that:

\sqrt{\left(-81\right)x^2}=9ix

7 0
2 years ago
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elena55 [62]
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7 0
3 years ago
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A
rjkz [21]

The other leg is 12cm long.

Step-by-step explanation:

Given,

Hypotenuse = c = 15cm

One leg = a = 9cm

Other leg = b

As the triangle is right angled, therefore, using Pythagoras theorem;

a^2+b^2=c^2\\(9)^2+b^2=(15)^2\\81+b^2=225\\b^2=225-81\\b^2=144\\

Taking square root on both sides;

\sqrt{b^2} =\sqrt{144} \\b=12\ cm

The other leg is 12cm long.

Keywords: triangle, square root

Learn more about triangles at:

  • brainly.com/question/5461619
  • brainly.com/question/5510873

#LearnwithBrainly

3 0
3 years ago
I don’t understand please help
Katarina [22]
Y= 4/1 x -9 :) djsjsjdndndndj
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3 years ago
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