Sin10+sin20+sin40+sin50=sin70+sin80.prove it

1 answer:

4 0

We are going to prove it like this:
Lets use the formula sinA+sinB=2sin(A+B/2)cos(A-B/2)
Now we are going to take the left side of equation
sin10+sin40+sin50+sin20
Arranging =(sin50+sin10)+(sin40+sin20)
Applying the above formula. =2sin(50+10/2)cos(50-10/2)+2sin(40+20/… =2sin(30)cos(20)+2sin(30)cos(10)
=2sin30{cos20+cos10}
Again using the formula
cosA+cosB= 2cos(A+B/2)cos(A-B/2)
=2sin30{2cos(20+10/2).cos(20-10/2)}
=2sin30{2cos(15).cos(5)}
=2(1/2){2cos15.cos5} as sin30=1/2
=2cos15.cos5
Taking right side of equation sin70+ sin80
Using the formula sinA+sinB
= 2sin(A+B/2)cos(A-B/2)
=2sin(70+80/2)cos(70-80/2)
=2sin75cos5
=2sin(90-15)cos5
=2cos15.cos5 <span>Hope this helps</span>

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Consider the following random sample from a normal population: 14, 10, 13, 16, 12, 18, 15, and 11. What is the 95% confidence in

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Answer:

$13.625-2.365\frac{2.669}{\sqrt{8}}=11.393$

$13.625+2.365\frac{2.669}{\sqrt{8}}=15.857$

So on this case the 95% confidence interval would be given by (11.393;15.857)

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Data: 14, 10, 13, 16, 12, 18, 15, 11

We can calculate the sample mean and deviation with the following formulas:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

$s = \sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}$

$\bar X=13.625$ represent the sample mean

$\mu$ population mean (variable of interest)

s=2.669 represent the sample standard deviation

n=8 represent the sample size

Calculate the confidence interval

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=8-1=7$

Since the confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,7)".And we see that $t_{\alpha/2}=2.365$