Prove-: sin5A = 5cos^4 A sinA - 10cos^2 A sin^3 A + sin^5 A

Answers

Answer 1

Answer: $A=x\n\nsin5x=5cos^4xsinx-10cos^2xsin^3x+sin^5x\n\nL=sin(3x+2x)=sin3xcos2x+sin2xcos3x=(*)\n-------------------------------\nsin3x=sin(2x+x)=sin2xcosx+sinxcos2x\n=2sinxcosxcosx+sinx(cos^2x-sin^2x)\n=2sinxcos^2x+sinxcos^2x-sin^3x=3sinxcos^2x-sin^3x\n-------------------------------$

$sin3xcos2x=(3sinxcos^2x-sin^3x)(cos^2x-sin^2x)\n=3sinxcos^4x-3sin^3xcos^2x-sin^3xcos^2x+sin^5x\n=3sinxcos^4x-4sin^3xcos^2x+sin^5x\n-------------------------------$

$cos3x=cos(2x+x)=cos2xcosx-sin2xsinx\n=cosx(cos^2x-sin^2x)-2sinxcosxsinx\n=cos^3x-sin^2xcosx-2sin^2xcosx=cos^3x-3sin^2xcosx\n-------------------------------$

$sin2xcos3x=2sinxcosx(cos^3x-3sin^2xcosx)\n=2sinxcos^4x-6sin^3xcos^2x\n-------------------------------$

$(*)=3sinxcos^4x-4sin^3xcos^2x+sin^5x+2sinxcos^4x-6sin^3xcos^2x\n\n=5sinxcos^4x-10sin^3xcos^2x+sin^5x=R$

$=======================================\n\nsin(\alpha+\beta)=sin\alpha cos\beta+sin\beta cos\alpha\n\ncos(\alpha+\beta)=cos\alpha cos\beta-sin\alpha sin\beta\n\nsin2\alpha=2sin\alpha cos\alpha\n\ncos\alpha=cos^2\alpha-sin^2\alpha$

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Katrina and Amanda are studying the same class. On Monday, Katrina solved 35 math problems and Amanda solved 21 math problems. On Wednesday Katrina solved 16 math problems and Amanda solved 20 math problems. On Thursday, Katrina solved 40 math problems and Amanda solved 24 math problems. On Saturday, Katrina solved 18 math problems and Amanda solved 24 math problems. On which day did Katrina and Amanda have the same ratio of problems as Monday?
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Determine whether the following problem involves a permutation or combination. (It is not necessary to solve the problem.) How many different 33-letter passwords can be formed from the letters Upper QQ, Upper RR, Upper SS, Upper TT, Upper UU, Upper VV, and Upper WW if no repetition of letters is allowed?

Answers

Answer:

Permutation

Step-by-step explanation:

Since we are talking about passwords the order of the letters matters, i.e, the pasword UpT is different to the pasword pUT. Therefore, the problem involves all the permutations of 33 letters, from the given letters, where no repetitions of letters are allowed.

PLEASE HELP! How do I solve this problem?

Answers

The toughest part of this problem could be deciding what names to give the quantities of one-Euro and two-Euro coins.

-- I called the number of one-Euro coins ' N '.
Each of them is worth 1 Euro, so all ' N ' of them are worth ' N ' Euros.

-- I called the number of two-Euro coins ' T '.
Each of them is worth 2 Euros, so all ' T ' of them are worth ' 2T ' Euros.

-- The total number of coins in Penny's pocket is (N + T), and it says there are 11.

-- Their total value is (N + 2T), and it says the total value is 18.

So now you have two equations, with two unknowns.

   N + 2T = 18
   N + T = 11

Subtract the bottom equation from the top one, and you get:

(N - N) + (2T - T) = (18 - 11)

   T = 7    (there are 7 two-Euro coins in her pocket)

That right there is the answer to the question, so you don't need to go
any farther. But if you wanted to, you could also figure out how many
one-Euro coins there are in her pocket.

While testing a new vaccine, a lab technician noticed that the number of bacteria reduced by half every minute. How long does it take for 3,000 bacteria to reduce to 375? It takes minutes for the number of bacteria to reduce from 3,000 to 375.

Answers

This is a geometric progression:
Tn=a₁r^(n-1)

In this case:
Tn=a₁r^n

Because:
T₁ corresponds to 0 minutes ⇒T₁=3000(1/2)⁰=3000
T₂ corresponds to 1 minute ⇒ T²=3000(1/2)¹=1500 ...

Tn=375
a₁=3000
r=1/2
n=number of minutes
Therefore:

375=3000*(1/2)^n
375/3000=(1/2)^n
0.125=(1/2)^n
Ln 0.125=ln [(1/2)^n]
Ln 0.125=n Ln(1/2)
Ln 0.125=n [Ln1-Ln2]
Ln 0.125=n[0-Ln 2]
n=Ln 0.125 / (-Ln2)
n=3

answer: 3 minutes.

Jacob’s recipe calls for 4 7/8 cups of raisins. He has 6/11 of the amount that he needs.About how many cups of raisins does Jacob have?

Estimate by first rounding each number to the nearest 1/2.

A.4 cups

B.5 1/2

C.2 1/2

D.3 1/2

Answers

If you would like to know how many cups of raisins does Jacob have, you can calculate this using the following steps:

6/11 ... 1/2
4 7/8 cups ... 5 cups

1/2 of 5 cups = 1/2 * 5 = 5/2 = 2 1/2 cups

The correct result would be C. 2 1/2.

What is 0.063 in scientific notation?

Answers

Remember that you must change this number so that it is between 1 and 10, and then multiply it by a power of 10.

So first, you have to move the decimal point over 2 places to the right.

Since you are moving to the right, the exponent on ten will be negative; since you moved 2 decimal places, you know the exponent will be -2.

The answer is 6.3 x 10^-2.

Hope this helps!

To put a number into scientific notation, we multiply or divide the number by 10 over and over until we have a number between 1 and 10. But we can't just change a number so we have to account for what we did by describing how many times we multiplied or divided with exponents.

Each time we multiply by 10 the decimal moves 1 place to the right. Each time we divide by 10 the decimal moves 1 place to the left. The exponent we use will describe the number of times we moved the decimal to the right or left. The exponent will be positive or negative to describe the direction we need to move the decimal to represent our original number.

I know... that sounds confusing, but it's really easy! Let's do yours...

0.063

Now we need to move the decimal to get a number between 1 and 10. So we move the decimal 2 places to the right to get 6.3

Now we have to describe how to get to the original number with the exponents. We moved the decimal 2 places right so to get our number back we would have to move it back 2 places to the left so negative exponent 2... Here's how we represent it..

6.3 x 10^-2

So what that says is if we take 6.3 and divide it by 10 twice we will get our original number.

Let's do another...

0.000000000623

We move the decimal 10 places to the right so our scientific notation is

6.23 x 10^-10

What about very large numbers...

650,000,000,000 = 6.5 x 10^11

(In that one the exponent is positive because we have to multiply to get back to the original number)

Hope this helps!

Which equation is an identity?A. 7m – 2 = 8m + 4 – m

B. 8y + 9 = 8y – 3

C. 11 – (2v + 3) = –2v – 8

D. 5w + 8 – w = 6w – 2(w – 4)

Answers

An identity is obtained when you substitute any value of x to an equation, both sides will give out the same answer. A trial-and-error method is best used for questions like these. The method is shown below:

Let: 1 = m = y = v = w

A. 7m – 2 = 8m + 4 – m

7(1) – 2 = 8(1) + 4 – (1)
5 =/= 11

B. 8y + 9 = 8y – 3

8(1) + 9 = 8(1) - 3
17 =/= 5

C. 11 – (2v + 3) = –2v – 8

11 - (2(1) + 3) = -2(1) - 8
6 =/= -10

D. 5w + 8 – w = 6w – 2(w – 4)

5(1) + 8 -1 = 6(1) - 2(1 - 4)
12 = 12

Therefore, the choice D is the identity.

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