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3 years ago

If cos theta + sin theta = root 2 cos thetha prove that cos theta - sin theta = root 2 sin theta

Mathematics

1 answer:

artcher [175]3 years ago

3 0

Answer:

We have:

If:

Cos(θ) + Sin(θ) = √2*cos(θ)

We want to prove that:

Cos(θ) - Sin(θ) = √2*Sin(θ)

Well, let's start with the first relation:

Cos(θ) + Sin(θ) = √2*cos(θ)

Now we can subtract 2*Sin(θ) and we will get:

Cos(θ) + Sin(θ) - 2*Sin(θ) = √2*cos(θ) - 2*sin(θ)

Cos(θ) - Sin(θ) = √2*cos(θ) - 2*sin(θ)

Now, we also can rewrite the first equation as:

Cos(θ) + Sin(θ) = √2*cos(θ)

Cos(θ) - √2*cos(θ) = - Sin(θ)

Cos(θ)*( 1 - √2) = -Sin(θ)

Cos(θ) = -Sin(θ)/( 1 - √2) = Sin(θ)/(- 1 + √2)

We can replace this in the right side of theequation:

Cos(θ) - Sin(θ) = √2*cos(θ) - 2*sin(θ)

Cos(θ) - Sin(θ) = √2* Sin(θ)/(- 1 + √2) - 2*sin(θ)

Cos(θ) - Sin(θ) = (√2/(- 1 + √2) - 2)*Sin(θ)

Now we have this:

√2/(- 1 + √2)) - 2 = a

if we multiply all by (-1 + √2) we get:

√2 - 2*(-1 + √2) = a*(-1 + √2)

√2 + 2 - 2*√2 = a*(-1 + √2)

2 - √2 = a*(-1 + √2)

√2*(√2 - 1) = a*(-1 + √2)

√2*(√2 - 1)/(-1 + √2) = a

√2 = a

Then:

√2/(- 1 + √2)) - 2 = √2

If we replace this in the equation:

Cos(θ) - Sin(θ) = (√2/(- 1 + √2) - 2)*Sin(θ)

We get:

Cos(θ) - Sin(θ) = √2*Sin(θ)

Which is the thing we wanted to get.

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3 years ago

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Step-by-step explanation:

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The speeds of vehicles traveling on a highway are normally distributed with an unkown population mean and standard deviation. A

Maksim231197 [3]

Answer:

The 90% confidence interval would be given by (60.09;69.91)

We are 90% confident that the true mean for the speeds of vehicles traveling on a highway is between 60.09 and 69.91 miles per hour.

Step-by-step explanation:

1) 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".

$\bar X =65$ represent the sample mean for the sample

$\mu$ population mean (variable of interest)

s=9 represent the sample standard deviation

n=11 represent the sample size

2) 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:

$df=n-1=11-1=10$

Since the Confidence is 0.90 or 90%, the value of $\alpha=0.1$ and $\alpha/2 =0.05$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,10)".And we see that $t_{\alpha/2}=1.81$

Now we have everything in order to replace into formula (1):

$65-1.81\frac{9}{\sqrt{11}}=60.09$

$65+1.81\frac{9}{\sqrt{11}}=69.91$

So on this case the 90% confidence interval would be given by (60.09;69.91)

We are 90% confident that the true mean for the speeds of vehicles traveling on a highway is between 60.09 and 69.91 miles per hour.

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3 years ago

15. The dollar amount y that remains to be repaid on a loan x months after the loan is issued is represented by y=-1000x + 8000.

irina [24]

Answer:

1) $8000

2) $1000

3) 8 months, since y represents our remaining amount to be paid, we set it equal to 0, to see when $0 need to be paid. Solving for x (months), we can it to be 8.

Step-by-step explanation:

We have the equation y = -1000x + 8000 which follows the linear equation:

y = mx + b, where m is our slope and b is our y-intercept

1) The initial balance can be found with our constant "b" which in this case is 8000. You can also plot the function of y and you will find that 8000 is the intercept when x = 0, aka the start

2) We can calculate the rate of change for when the loan is repaid by looking at the slope "m", in this case it is 1000. It subtracts 1000 each month, meaning $1000 is being payed and taken out of the bank account

3) To find how many months it will take for the loan to be repaid, let's solve for x when y = 0.

0 = -1000x + 8000

-8000 = -1000x

8 = x

It will take 8 months. Why? Since y represents our remaining amount to be paid, we set it = 0, to see when $0 need to be paid. Solving for x (months), we can it to be 8.

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