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Fudgin [204]
3 years ago
7

Triangle MNP is the image of triangle JKL after a 120 degrees counterclockwise rotation about point Q. If the measure of angle L

is 47 degrees and the measure of angle N is 57 degrees, determine the measure of angle M. Explain how you arrived at your answer.

Mathematics
1 answer:
gtnhenbr [62]3 years ago
5 0
Either 129 or 46 inches
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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
2 years ago
Three less than twice a number is 15.
inessss [21]

Answer:

3-2x=15

Step-by-step explanation:

add 3 to both sides makes -2x=18. divide -2 from both side gets the answer x=-9, so the answer is C, -9

5 0
3 years ago
Dan has 3 ⅔ cups of milk. If he used ¼ of the milk for his breakfast, how much milk does Dan have left?
riadik2000 [5.3K]

There would be 3 2/12 left.

2/3 x 1/4 = 2/12

Hope this helps!

Sorryyyy :C

6 0
3 years ago
Read 2 more answers
The body of a person who weighs 50 kilograms contains around 2 × 102 grams of sodium chloride. How much sodium chloride is conta
Zolol [24]
I'm not sure but I.think it's C
4 0
3 years ago
What is the value of x that makes the given equation true 9+8x-4-5x=-19
Elena L [17]

Answer:

x = -8

Step-by-step explanation:

9+8x-4-5x=-19

5+8x-5x=-19

5+3x=-19

-5 =-19

3x = -24

3/3 = -24/3

x= -8

8 0
2 years ago
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