Prove that $cos 3 x =4 cos ^(3)x-3 cos x$

Answers

Answer 1

Answer: $cos3x=4cos^3x-3cosx\n\nL=cos(2x+x)=cos2xcosx-sin2xsinx\n\n=(cos^2x-sin^2x)cosx-2sinxcosxsinx\n\n=cos^3x-sin^2xcosx-2sin^2xcosx\n\n=cos^3x-3sin^2xcosx\n\n=cos^3x-3cosx(1-cos^2x)\n\n=cos^3x-3cosx+3cos^3x\n\n=4cos^3x-3cosx=R$

$cos(a+b)=cosa\ cosb-sina\ sinb\n\nsin^2a+cos^2b=1\to sin^2a=1-cos^2b\n\ncos2a=cos^2a-sin^2a\n\nsin2a=2sina\ cosa$

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What is the relationship among proportional relationships lines, rates of change, and slope?

Answers

First of all, let's define some concepts. In mathematics, proportional relationships happen when two values always change by the same multiple. That is, when one doubles, the other doubles as well. So, you can always reduce a proportional relationship to the same equation as follows:

$y=kx \n where \ k \ is \ the \ proportionality \ constant$

This constant is also the slope of the straight line.

On the other hand, the average rate of change between any two points $(x_(1),f(x_(1)) \ and \ (x_(2),f(x_(2))$ is the slope of the line through the two points. The line through the two points is called the secant line and its slope is denoted as $m_(sec)$ , so:

$ARC=(f(x_(2))-f(x_(1)))/(x_(2)-x_(1)) \n \n \therefore ARC=(Change \ in \ y)/(Change \ in \ x)=m_(sec)$

Finally, The slope of a nonvertical line is the number of units the line rises (or falls)vertically for each unit of horizontal change from left to right, so:

$Slope=(change \ in \ y)/(change \ in \ x)=(rise)/(run)$

So, we can say that the relationship of these three concepts is that they all talk about the slope of a curve.

It introduces the relationship between two variables and is called correlation. Proportionality or variation is state of relationship or correlation between two variables It has two types: direct variation or proportion which states both variables are positively correlation. It is when both the variables increase or decrease together. On the contrary, indirect variation or proportion indicates negative relationship or correlation. Elaborately, the opposite of what happens to direct variation. One increases with the other variables, you got it, decreases. This correlations are important to consider because you can determine and identify how two variables relates with one another. Notice x = y (direct), y=1/x (indirect)

How to find an equation of line parallel to a line?

Answers

If two lines are parallel, then they have the same slope. Take the slope
of the line that you know, and use it for the new line. If you also need to
find the intercept of the new line, then you need some more information ...
the intercept is not the same as the line that you know.

Final answer:

To find the equation of a line parallel to a given line, identify the slope of the given line and choose any point on that line, then use the point-slope formula to find the equation of the parallel line.

Explanation:

To find the equation of a line parallel to a given line, follow these steps:

Identify the slope of the given line. Let's call it 'm'.
Select any point on the given line. Let's call the coordinates of this point (x₁, y₁).
Use the point-slope formula, y - y₁ = m(x - x₁), where (x, y) represents any point on the parallel line.
Simplify the equation to get the final equation of the line parallel to the given line.

Let me know if you need further assistance!

Learn more about Equation of a parallel line here:

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What would you have to do to change 10 cubic feet into cubic inches? A. Multiply by 46,656 B. Divide by 1,728 C. Multiply by 1,728 D. Divide by 46,656

Answers

1 foot = 12 inches

(1 foot)³ = (12 inches)³

12³ = 1,728 in³

To change 10 cubic feet into cubic inches

10 ft³ * 1728 in³/ft³ = 17,280 in³ Choice C. Multiply by 1,728

C. Multiply by 1,728 is the answer

What is the missing value? The Volume is 5120 m^3

Answers

I think that the missing value is 16m.

I just multiplied two other numbers with 20 and 20 x 16 x 16 = 5120

I hope this was right.

Answer:

Step-by-step explanation:

V= $\pi$ r^2

5120= $\pi r^(2)$ x20

5120/20 = 256

$√(256)$ =16

16+16 ( 16x2) = 32

32in.

A pet shelter houses 32 cats and 45 dogs. What is the ratio of dogs to total number of cats and dogs? (Write your answer using the a:b format. For example, 1:1.)

Answers

Answer: 32:45

Step-by-step explanation: They share no common factors so it would just be 32:45

the answer is 45:77 . the question says “ ratio of dogs ( 45 ) “ to “ the total number of cats AND dogs “ which means cats and dogs combined which is 77/

Find the proportion please answer number 5. 6. 8. And 9. :)

Answers

Answer:

5. 28

6. 6

8. 1,232

9. 2

Step-by-step explanation:

When you know both numerators (the numbers on top) or both denominators (numbers on bottom), you can divide the second by the first and then apply that to the other side. For example, on question five, I divided twenty by fifteen. The answer is 1.33333... or $1(1)/(3)$ , or $(4)/(3)$ . So I multiply 21 by this number and my answer is 28. If visualization helps:

$(15)/(21) =(20)/(y)$

Step 1: 20 ÷ 15 = $(4)/(3)$ (second number divided by first number, using the side where we know both numbers)

Step 2: 21 × $(4)/(3)$ = 28 (y = 28. solved!)

You can do the same thing on question nine, even though the known numbers are on the bottom instead of the top. Fifteen divided by 120, then multiply sixteen by your answer - 0.125 - and you will see that j (i?) = 2 :)

If you have any questions, feel free to ask. Hope this helped!

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Phil and Nick are selling fruit for a fundraiser. Customers can buy small and large boxes of tangerines. Phil sold 11 small boxes and 10 large boxes of tangerines for a total of $146. Nick sold 5 small boxes of tangerines and 12 large boxes of tangerines for a total of $126. Write the systems of equations. What is the cost of a large box of tangerines?