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N76 [4]
3 years ago
13

A. Show

Mathematics
1 answer:
nydimaria [60]3 years ago
8 0

a. Recall the double angle identities:

\sin^2x=\dfrac{1-\cos2x}2

\cos^2x=\dfrac{1+\cos2x}2

Then

\sin^2x\cos^2x=\dfrac{(1-\cos2x)(1+\cos2x)}4=\dfrac{1-\cos^22x}4

Applying the identity again, we have

\sin^2x\cos^2x=\dfrac{1-\frac{1+\cos4x}2}4=\dfrac{2-(1+\cos4x)}8=\dfrac{1-\cos4x}8

as required.

b. Using the result from part (a),

\sin^2x\cos^2x=\dfrac{1-\cos4x}8=\dfrac{2-\sqrt2}{16}

\implies\cos4x=\dfrac1{\sqrt2}

\implies4x=\dfrac\pi4+2n\pi\text{ or }4x=-\dfrac\pi4+2n\pi

(where n is any integer)

\implies\boxed{x=\pm\dfrac\pi{16}+\dfrac{n\pi}2}

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3 years ago
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Farmer Smith enclosed his rectangular pasture with fence that cost him $1.50 per linear meter. If his pasture is twice as long a
maria [59]

Answer:

$2,700

Step-by-step explanation:

Let the width of the pasture be x, this means that the length of the pasture will be 2x

The area of a rectangle is L * B

Hence,

x * 2x = 18,000

2x^2 = 180,000

x^2 = 90000

x = 300 meters

This means the length is 600 metres

Now, to get the length of the fence, we need to know the actual perimeter which is 2(L + B)

= 2 ( 300 + 600)

2 * 900 = 1,800 metres

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6 0
3 years ago
PLEASE HELP ME PICTURES INCLUDED
Nataliya [291]

Two equations given. We have to find the solution sets for the equations.

y = -x²-2x+8 and y = 2x +11

There are 3 steps given. As both equations are equal to y, so we can equate the x parts.

-x²-2x+8 = 2x+11

0 = x²+4x+3

0 = (x+1)(x+3)

Now we have to use zero product property to solve this. If multiplication of two terms equal to 0, we can set each of them equal to 0.

x+1 = 0 , x+3 = 0

When, x+1 =0, by moving 1 to the other side we will get x = -1.

When x+3 =0, by moving 3 to the other side we will get x =-3

Now for the two values of x, we have to find y .

When x = -1, by plugging in the value to the second equation we will get,

y = 2x+11 = 2×(-1)+11 = -2 +11 = 9

So for x =-1, y =9.

The solution set is (-1, 9)

Now when x = -3, by plugging in the value to the second equation we will get,

y = 2x+11 = 2×(-3) +11 = -6+11 = 5

So for x = -3, y = 5

The solution set is (-3,5)

Therefore, the solutions for the system of equations are (-3, 5) and (-1, 9)

So the third option is correct.

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3 years ago
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DaniilM [7]

Answer:3

Step-by-step explanation:

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3 years ago
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Anastaziya [24]

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