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2 years ago

Triangle LPQ has side lengths of 9, 20, and 15. Classify the triangle by its sides

Mathematics

1 answer:

Aneli [31]2 years ago

5 0

Since all of the side lengths are different, it is a scalene triangle

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<img src="https://tex.z-dn.net/?f=%20%5Crm%20%5Cint_%7B0%7D%5E%7B%20%20%5Cpi%20%7D%20%5Ccos%28%20%5Ccot%28x%29%20%20%20%20-%20%2

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

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

7 0

1 year ago

How to graph -x + 2y = 6

larisa86 [58]

In order to graph the equation, you need some coordinates,
So, put x = 0,
2y = 6. y = 3. Coordinate = (0, 3)

Now, put y = 0,
-x = 6
x = -6 Coordinate (-6, 0)

In short, Just mark that two points and draw a line, graph is done

Hope this helps!

3 0

2 years ago

Laura’s printer takes 5 minutes to print 85 pages.

evablogger [386]

Answer:

17.

Step-by-step explanation:

<u>Divide 85 by 5 to find the unit rate:</u>

<u /> $\frac{85}{5} =17$

17 represents the amount of pages printed for 1 minute.

5 0

2 years ago

Read 2 more answers

William and Emma go into a burger shop with a total of $40.00 to spend. Burgers cost $4.50 each and soft drinks cost $1.00 each.

Ainat [17]

Answer:

12 items in total

Step-by-step explanation:

40-4.5*8=4
leaving 4 dollars to spend on 4 drinks

5 0

2 years ago

Hannah has $400 to spend on vacation. She is going to be gone for 3 days. What is the most she

eimsori [14]

Answer:

I'll spend about maybe $80

Step-by-step explanation:

So I can have fun also this is a joke not real answer >:D

8 0

2 years ago