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4 years ago

Explain how the pythagorean theorem is used to derive the distance formula

Mathematics

1 answer:

kenny6666 [7]4 years ago

3 0

Answer:

In the Pythagorean Theorem, a right triangles hypotenuse is found by a^2 +b^2= c^2. The distance formula is basically the same thing. With two points, a right triangle can be formed and the slop would be the hypotenuse. That's where the distance formula comes from

Step-by-step explanation:

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2.48571428571

Step-by-step explanation:

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3 years ago

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3 years ago

Find the value of x<br>a) 5<br>b) 4<br>c) 6<br>d) 7

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Nikolay [14]

Replace x with π/2 - x to get the equivalent integral

$\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx$

but the integrand is even, so this is really just

$\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx$

Substitute x = 1/2 arccot(u/2), which transforms the integral to

$\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du$

There are lots of ways to compute this. What I did was to consider the complex contour integral

$\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz$

where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be

$\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}$

which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit

$\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}$

and it follows that

$\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}$

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2 years ago

Last year, Mr. Jones made $30,000. His boss just informed him that he will be receiving at least an 11.2% raise for this year. H

Lady bird [3.3K]

Answer:

$33 336

Step-by-step explanation:

Increase = 30 000 × 0.112

Increase = $3336

New salary = 30 000 + 3336

New salary = $33 336

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3 years ago

Explain how the pythagorean theorem is used to derive the distance formula​

Explain how the pythagorean theorem is used to derive the distance formula