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Lynna [10]
3 years ago
8

What is the reciprocal of 9/11

Mathematics
2 answers:
SVETLANKA909090 [29]3 years ago
7 0
The reciprocal is the flipped value so it will be11/9
Tpy6a [65]3 years ago
7 0
 your answer will be11/9.
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Question (c)! How do I know that t^5-10t^3+5t=0?<br> Thanks!
astra-53 [7]
(a) By DeMoivre's theorem, we have

(\cos\theta+i\sin\theta)^5=\cos5\theta+i\sin5\theta

On the LHS, expanding yields

\cos^5\theta+5i\cos^4\theta\sin\theta-10\cos^3\theta\sin^2\theta-10i\cos^2\theta\sin^3\theta+5\cos\theta\sin^4\theta+i\sin^4\theta

Matching up real and imaginary parts, we have for (i) and (ii),


\cos5\theta=\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta
\sin5\theta=5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta

(b) By the definition of the tangent function,

\tan5\theta=\dfrac{\sin5\theta}{\cos5\theta}
=\dfrac{5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta}{\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta}

=\dfrac{5\tan\theta-10\tan^3\theta+\tan^5\theta}{1-10\tan^2\theta+5\tan^4\theta}
=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}


(c) Setting \theta=\dfrac\pi5, we have t=\tan\dfrac\pi5 and \tan5\left(\dfrac\pi5\right)=\tan\pi=0. So

0=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}

At the given value of t, the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.


0=t^5-10t^3+5t\implies0=t^4-10t^2+5

Remember, this is saying that

0=\tan^4\dfrac\pi5-10\tan^2\dfrac\pi5+5

If we replace \tan^2\dfrac\pi5 with a variable x, then the above means \tan^2\dfrac\pi5 is a root to the quadratic equation,

x^2-10x+5=0

Also, if \theta=\dfrac{2\pi}5, then t=\tan\dfrac{2\pi}5 and \tan5\left(\dfrac{2\pi}5\right)=\tan2\pi=0. So by a similar argument as above, we deduce that \tan^2\dfrac{2\pi}5 is also a root to the quadratic equation above.

(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write

x^2-10x+5=\left(x-\tan^2\dfrac\pi5\right)\left(x-\tan^2\dfrac{2\pi}5\right)

Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy

5=\tan^2\dfrac\pi5\tan^2\dfrac{2\pi}5\implies\tan\dfrac\pi5\tan\dfrac{2\pi}5=\pm\sqrt5

But \tanx>0 for all 0, as is the case for x=\dfrac\pi5 and x=\dfrac{2\pi}5, so we choose the positive root.
3 0
3 years ago
What is the value of x in the equation 3x-4y=65 when y =4
Dovator [93]
3x - 4(4) = 65

First, simplify 4 × 4 to get 16. / Your problem should look like: 3x - 16 = 65
Second, add 16 to both sides. / Your problem should look like: 3x = 65 + 16
Third, simplify 65 + 16 to 81. / Your problem should look like: 3x = 81
Fourth, divide both sides by 3. / Your problem should look like: x = \frac{81}{3}
Fifth, simplify \frac{81}{3} to 27. / Your problem should look like: x = 27

Answer: x = 27

5 0
3 years ago
Write the equation of the line in
Brut [27]

Answer:

y - 3= 2(x - 0)  

Step-by-step explanation:

hello :

note 1 :

the point-slope formula is :

y - y_1 = m(x - x_1)   the point is A(x_1  , y_1 )  , m :  the slope

note 2 :

the slope is :   m = (y_2 - y_1)/(x_2 -x_1)  when given two points  :

A(x_1  , y_1 )  and  B(x_2  , y_2 )

in this exercice :  x_1 =0      y_1 =3            x_2 =1      y_2 =1

calculate now   m = (1 - 3)/(1 -0) =-2

an equation  in point-slope form is : y - y_1 = m(x - x_1)  

y - 3= 2(x - 0)  

5 0
3 years ago
What is the opposite of the opposite of any integer?​
Pani-rosa [81]
<h2>The opposite of an opposite of a integer is the original integer.</h2>
8 0
2 years ago
Read 2 more answers
A fruit seller purchased 20 kg of apples at Rs 80 per kg and sold them at Rs 85 per kg. find profit and profit percent​
viktelen [127]

Answer:

Profit = Rs 100

Profit %age = 6.25%

Step-by-step explanation:

Cost Price = Rs 80

Cost Price for 20 kg = Rs 1600

Selling Price = Rs 85

<u><em>Profit for 1 kg</em></u> = Rs 85 - Rs 80

=> Rs 5 (For 1 Kg)

<u><em>Profit for 20 kg:</em></u>

=> Rs 5*20

=> Rs 100

<u><em>Now Profit %age:</em></u>

=> \frac{100}{1600}* 100

=> 0.0625 * 100

=> 6.25%

7 0
3 years ago
Read 2 more answers
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