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Natalka [10]
3 years ago
8

Find all solutions to the equation. cos2x + 2 cos x + 1 = 0

Mathematics
2 answers:
STALIN [3.7K]3 years ago
3 0
First of all, we must use the equivalent formula of cos2x, it is cos2x=2cos^2x-1,
the equation has become 2cos^2x-1 +2cosx +1 = 0, it means 2cos^2x+2cosx = 0, or
cosx(cosx+1)=0, we get cosx=0, implying x=Pi/2, or (cosx+1)=0,  cosx= -1
implying x = Pi
for x=0, cos2*0+2cos0+1=1+2+1=4 impossible
for x= Pi, cos2*Pi+2cosPi+1=1-2+1=0 it is exact , so the solution is x=Pi
Genrish500 [490]3 years ago
3 0

Answer:

Here, the given expression is,

cos(2x)+2cosx+1=0

\implies (2cos^2x-1)+2cosx+1=0  ( Because cos(2x) = 2cos²x-1 )

\implies 2cos^2x+2cosx=0

\implies 2cosx(cosx+1)=0

Thus, 2 cos x = 0 or cos x + 1 = 0

If ,

2cos x = 0

⇒ cos x = 0

⇒ x = \frac{pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}.......

⇒ x = \frac{\pi}{2}+n\pi

Where, n = 0, 1, 2, ......

Now, if,

cos x = -1

⇒ x=\pi, 3\pi, 5\pi,.....

⇒ x = \pi+n\pi

Where, n = 0, 1, 2, ......

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son4ous [18]
There is a relationship between confidence interval and standard deviation:
\theta=\overline{x} \pm \frac{z\sigma}{\sqrt{n}}
Where \overline{x} is the mean, \sigma is standard deviation, and n is number of data points.
Every confidence interval has associated z value. This can be found online.
We need to find the standard deviation first: 
\sigma=\sqrt{\frac{\sum(x-\overline{x})^2}{n}
When we do all the calculations we find that:
\overline{x}=123.8\\ \sigma=11.84
Now we can find confidence intervals:
($90\%,z=1.645): \theta=123.8 \pm \frac{1.645\cdot 11.84}{\sqrt{15}}=123.8 \pm5.0\\($95\%,z=1.960): \theta=123.8 \pm \frac{1.960\cdot 11.84}{\sqrt{15}}=123.8 \pm 5.99\\ ($99\%,z=2.576): \theta=123.8 \pm \frac{2.576\cdot 11.84}{\sqrt{15}}=123.8 \pm 7.87\\
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Here is the link to the spreadsheet with standard deviation calculation:
https://docs.google.com/spreadsheets/d/1pnsJIrM_lmQKAGRJvduiHzjg9mYvLgpsCqCoGYvR5Us/edit?usp=sharing
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20liters of mixture contain milk nad water in the ratio 5:3 if 4liters of the mixture are replaced by 4 liters of milk find the
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Answer:

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Step-by-step explanation:

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the amount of milk in 16 liter=16*5/8=10 liter of milk

the amount of water in 16 liter=16*3/8= 6 liter

add 4 liter of milk to the mixture: 10+4=14 liter of milk and 6 liter of water

the new ratio milk/water is  14:6  or 7:3

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