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zavuch27 [327]
3 years ago
10

What are the solutions to the equation Sine (x + StartFraction 7 pi Over 2 EndFraction) = negative StartFraction StartRoot 3 End

Root Over 2 EndFraction over the interval [0, 2Pi]?
Mathematics
2 answers:
Blababa [14]3 years ago
5 0

Answer:

C. π/6 & 11π/6

Step-by-step explanation:

If you graph the equation ( Sin (x+7π/2)=-√3/2) and look between 0 & 2π, you'll see that the lines intersect the x-axis at π/6 & 11π/6.

il63 [147K]3 years ago
4 0

Given:

The equation is

\sin\left(x+\dfrac{7\pi}{2}\right)=-\dfrac{\sqrt{3}}{2}

To find:

The solutions of given equation over the interval [0,2\pi].

Solution:

We have,

The equation is

\sin\left(x+\dfrac{7\pi}{2}\right)=-\dfrac{\sqrt{3}}{2}

\sin\left(x+\dfrac{7\pi}{2}\right)=-\sin \dfrac{\pi }{3}

\sin\left(x+\dfrac{7\pi}{2}\right)=\sin (-\dfrac{\pi }{3})

If \sin x=\sin y, then x=n\pi +(-1)^ny.

Over the interval [0,2\pi].

x+\dfrac{7\pi}{2}=4\pi-\dfrac{\pi }{3} and x+\dfrac{7\pi}{2}=5\pi+\dfrac{\pi }{3}

x=\dfrac{11\pi }{3}-\dfrac{7\pi}{2} and x=\dfrac{16\pi}{3}-\dfrac{7\pi}{2}

x=\dfrac{22\pi-21\pi }{6} and x=\dfrac{32\pi-21\pi }{6}

x=\dfrac{\pi}{6} and x=\dfrac{11\pi }{6}

Therefore, the two solutions are tex]x=\dfrac{\pi}{6}[/tex] and x=\dfrac{11\pi }{6}.

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siniylev [52]

Answer:

\dfrac{AP}{PC}=\dfrac{3}{5}

\dfrac{BN}{CN}=\dfrac{3}{5}

Step-by-step explanation:

Consider triangles AMP and ADC. In these triangles,

  • angle A is the common angle, so \angle MAP\cong \angle DAC by reflexive property;
  • angles AMP and ADC are congruent as corresponding angles when two parallel lines MP and CD are cut by transversal AD.

Hence, triangles AMP and ADC are similar by AA similarity theorem.

Similar triangles have proportional corresponding sides, thus

\dfrac{AM}{AD}=\dfrac{AP}{AC}\\ \\\dfrac{3x}{3x+5x}=\dfrac{AP}{AC}\\ \\\dfrac{AP}{AC}=\dfrac{3}{8}\Rightarrow AP=\dfrac{3}{8}AC\\ \\PC=AC-AP=AC-\dfrac{3}{8}AC=\dfrac{5}{8}AC,

so

\dfrac{AP}{PC}=\dfrac{\frac{3}{8}AC}{\frac{5}{8}AC}=\dfrac{3}{5}

Consider triangles ACB and PCN. In these triangles,

  • angle C is the common angle, so \angle ACB\cong \angle PCN by reflexive property;
  • angles ABC and PCN are congruent as corresponding angles when two parallel lines PN and AB are cut by transversal BC.

Hence, triangles ACB and PCN are similar by AA similarity theorem.

Similar triangles have proportional corresponding sides, thus

\dfrac{CP}{AP}=\dfrac{CN}{CB}\\ \\\dfrac{5x}{3x+5x}=\dfrac{CN}{CB}\\ \\\dfrac{CN}{CB}=\dfrac{5}{8}\Rightarrow CN=\dfrac{5}{8}CB\\ \\BN=BC-CN=BC-\dfrac{5}{8}BC=\dfrac{3}{8}BC,

so

\dfrac{BN}{CN}=\dfrac{\frac{3}{8}BC}{\frac{5}{8}BC}=\dfrac{3}{5}

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