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never [62]
4 years ago
10

Sin(x) + sin(3x) = 4 sin(x) cos^2(x) Prove the identity.

Mathematics
1 answer:
salantis [7]4 years ago
6 0

\bf \textit{Sum to Product Identities} \\\\ sin(\alpha)+sin(\beta)=2sin\left(\cfrac{\alpha+\beta}{2}\right)cos\left(\cfrac{\alpha-\beta}{2}\right)\leftarrow \textit{we'll use this one} \\\\\\ sin(\alpha)-sin(\beta)=2cos\left(\cfrac{\alpha+\beta}{2}\right)sin\left(\cfrac{\alpha-\beta}{2}\right) \\\\\\ \stackrel{\textit{symmetry identity}}{cos(-\theta )=cos(\theta )} \\\\[-0.35em] \rule{34em}{0.25pt}

\bf sin(x)+sin(3x)=\underline{4sin(x)cos^2(x)} \\\\[-0.35em] ~\dotfill\\\\ sin(x)+sin(3x)\implies 2sin\left( \cfrac{x+3x}{2} \right)cos\left( \cfrac{x-3x}{2} \right) \\\\\\ 2sin\left( \cfrac{4x}{2} \right)cos\left( \cfrac{-2x}{2} \right)\implies 2sin(2x)cos(-x)\implies 2\boxed{sin(2x)} cos(x) \\\\\\ 2\boxed{2sin(x)cos(x)} cos(x)\implies \underline{4sin(x)cos^2(x)}

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g100num [7]
The answer is 5.196152423

Since - x^2=27
X=square root of 27
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4 0
3 years ago
Penny needs 30 cubic ft of sand to fill a sandbox. She plans to buy 10 bags of sand for her sandbox. Each bag contains 2.9 cubic
Romashka [77]
The correct answer is C.

Because: 2.9 multiplied by 10 bags equals 29cubic ft and she needs 30 cubic ft. So she would need to buy 11 bags to make sure she has enough sand for the whole thing. 11 x 2.9 = 31.9 which ensures she has enough for the whole thing including extra in case. So you answer would be Answer C.
4 0
3 years ago
5x^2-2x=2<br>help plsmmm​
Yuki888 [10]

Answer:

Presumably you're solving for x here?  Without further information we'll assume that.

With that in mind, x is approximately equal to 0.86 and -0.46

Step-by-step explanation:

Let's start by putting it in the usual ax² + bx + c format.

5x^2- 2x = 2\\5x^2 - 2x - 2 = 0\\

let's solve it.  First we'll multiply both sides by five, making the first term a perfect square:

25x^2 - 10x - 10 = 0\\

Now we'll add 11 to both sides:

25x^2 - 10x + 1 = 11\\

Which makes the left side a perfect square:

(5x - 1)^2 = 11

And now we can solve for x:

(5x - 1)^2 = 11\\\\5x - 1 =  \sqrt{11} \\\\5x = 1 + \sqrt{11}\\\\x = \frac{1 + \sqrt{11}}{5}

Note that there's no apparent way of drawing the ± symbol when editing equations, so take that + sign as actually being ±.

That gives us two answers:

x = \frac{1 + \sqrt{11}}{5}\\\\x \approx \frac{1 + 3.32}5\\\\x \approx \frac{4.32}5\\\\x \approx 0.86             x = \frac{1 - \sqrt{11}}{5}\\\\x \approx \frac{1 - 3.32}5\\\\x \approx \frac{-2.32}5\\\\x \approx -0.46

7 0
3 years ago
3 sisters have ages that are consecutive odd integers. The sum of the age of the youngest and three times the age of the oldest
Naddika [18.5K]

Step-by-step explanation:

Let n be the age of the youngest sister. Since the ages are consecutive odd integers, the next sister's age is n+2, and the oldest sister is n+4. the following equation represents the relationship between the girls' ages.

n + 3(n + 4) = 5(n + 2) - 5

n + 3n + 12 = 5n + 10 - 5

4n + 12 = 5n +5

-4n  -5      -4n -5

7 = n

so the sisters are 7, 9, and 11

4 0
3 years ago
Jill can make $125 in 5 hours shoveling driveways and Cybil can make $66 in 2 hours shoveling driveways. Maintaining these wages
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Cybil would make more money, he would get $99 as Jill would only get $75. The difference of the two would be $24.
4 0
3 years ago
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