Mathematics: What are the solutions to the equation Sine (x + StartFraction 7 pi Over 2 EndFraction) = negative StartFraction StartRoot 3 EndRoot Over 2 EndFraction over the interval [0, 2Pi]?

Step-by-step explanation:Plotting the points, you can see that the base is 9 units and the height is 7. This matches only the last answer choice....

What are the solutions to the equation Sine (x +

roro2599

09.02.2022

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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