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3 years ago

Simplify the expression square root of -1 over (3+8i) - (2+5i)

1 answer:

8 0

$\frac{i}{(3 \: + \: 8i) \: - \: (2 \: + \: 5i)} \: = \: \frac{i}{1 \: + \: 3i} \: = \: \frac{i}{1 \: + \: 3 i} \: \times \: \frac{1 \: - \: 3i}{1 \: - \: 3i} \: = \: \frac{i \: + \: 3}{1 \: + \: 9} \: = \: \frac{3 \: + \: i}{10}$ A) (3 + i)/10

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Step-by-step explanation:

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Help me with trigonometry

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Answer:

See below

Step-by-step explanation:

It has something to do with the Weierstrass substitution, where we have

$\int\, f(\sin(x), \cos(x))dx = \int\, \dfrac{2}{1+t^2}f\left(\dfrac{2t}{1+t^2}, \dfrac{1-t^2}{1+t^2} \right)dt$

First, consider the double angle formula for tangent:

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Once the double angle identity for sine is

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we know $\sin(x)=\dfrac{2t}{1+t^2}$ , but sure, we can derive this formula considering the double angle identity

$\sin(x)= 2\sin\left(\dfrac{x}{2}\right)\cos\left(\dfrac{x}{2}\right)$

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$\sin \arctan t = \dfrac{t}{\sqrt{1 + t^2}} \text{ and } \cos \arctan t = \dfrac{1}{\sqrt{1 + t^2}}$

Thus,

$\sin(x)= 2 \left(\dfrac{t}{\sqrt{1 + t^2}}\right) \left(\dfrac{1}{\sqrt{1 + t^2}}\right) = \dfrac{2t}{1 + t^2}$

Similarly for cosine, consider the double angle identity

Thus,

$\cos(x)= \left(\dfrac{1}{\sqrt{1 + t^2}}\right)^2- \left(\dfrac{t}{\sqrt{1 + t^2}}\right)^2 = \dfrac{1}{t^2+1}-\dfrac{t^2}{t^2+1} =\dfrac{1-t^2}{1+t^2}$

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======================================================

$5\cos(x) =12\sin(x) +3, x \in [0, 2\pi ]$

Solving

$5\,\overbrace{\frac{1-t^2}{1+t^2}}^{\cos(x)} = 12\,\overbrace{\frac{2t}{1+t^2}}^{\sin(x)}+3$

$\implies \dfrac{5-5t^2}{1+t^2}= \dfrac{24t}{1+t^2}+3 \implies \dfrac{5-5t^2 -24t}{1+t^2}= 3$

$\implies 5-5t^2-24t=3\left(1+t^2\right) \implies -8t^2-24t+2=0$

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Just note that

$\tan\left(\dfrac{x}{2}\right) = \dfrac{3\pm 8\sqrt{10}}{-2}$

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3 years ago

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3 years ago

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Paha777 [63]

Answer:

Option1 : Domain and range for both functions are all real numbers

Step-by-step explanation:

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2 years ago

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Anna35 [415]

Answer: The fabric costs $7.60 per yard.

Step-by-step explanation: First we can divide it by 106 then multiply it by 100 to get rid of the 6% sales tax, which now the cost of the stuff is $17.50.

Next, we subtract 8 from this to represent the clothing pattern cost. Now it is $9.50.

The last stap to this is to divide it by 1 1/4 to get the final cost per yard:

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