Solve the equation in the interval from \dfrac{3\pi}{2} 2 3π start fraction, 3, pi, divided by, 2, end fraction too \dfrac{9\pi}{2} 2 9π start fraction, 9, pi, divided by, 2, end fraction. Your answer should be in radians. All choices are rounded to the nearest hundredth. \sin(x)=0.65sin(x)=0.65.

We have been given an equation [tex]\text{sin}(x)=0.65[/tex]. We are asked to solve the given equation in the interval [tex][\frac{3\pi}{2},\frac{9\pi}{2}][/tex].

Taking inverse of sine function, we will get:

[tex]x=\text{sin}^{-1}(0.65)[/tex]

General solutions of the equation would be:

[tex]x=\text{sin}^{-1}(0.65)+2\pi n,x=\pi -\text{sin}^{-1}(0.65)+2\pi n[/tex]

[tex]x=0.707584436725+2\pi n,x=\pi -0.707584436725+2\pi n[/tex]

Now, we need to find value of x such that:

[tex]\frac{3\pi}{2}\leq x\leq \frac{9\pi}{2} \text{ or } 4.71239\leq x\leq 14.13717[/tex]

When [tex]n=1[/tex]

[tex]x=0.707584436725+2\pi (1)\Rightarrow 0.707584436725+6.283185=6.990769436725\approx 6.99[/tex]

[tex]x=\pi -0.707584436725+2\pi (1)\Rightarrow 3.14159265-0.707584436725+6.283185307=8.717193520275\approx 8.72[/tex]

When [tex]n=2[/tex]:

[tex]x=0.707584436725+2\pi (2)\Rightarrow 0.707584436725+12.5663706=13.273955036725\approx 13.27[/tex]

[tex]x=\pi -0.707584436725+2\pi (2)\Rightarrow 3.14159265-0.707584436725+12.566370614=15.000378827275\approx 15.00[/tex]

We can see that [tex]x=15[/tex] is greater than [tex]14.137[/tex], therefore, the solutions for the given equation are [tex]x=6.99,8.72,13.27[/tex].