Question

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.

Answer 1

`x=6.99,8.72,13.27`.

Explanation:

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

Taking inverse of sine function, we will get:

`x=\text{sin}^(-1)(0.65)`

General solutions of the equation would be:

`x=\text{sin}^(-1)(0.65)+2\pi n,x=\pi -\text{sin}^(-1)(0.65)+2\pi n`

`x=0.707584436725+2\pi n,x=\pi -0.707584436725+2\pi n`

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

`(3\pi)/(2)\leq x\leq (9\pi)/(2) \text{ or } 4.71239\leq x\leq 14.13717`

When
`n=1`

`x=0.707584436725+2\pi (1)\Rightarrow 0.707584436725+6.283185=6.990769436725\approx 6.99`

`x=\pi -0.707584436725+2\pi (1)\Rightarrow 3.14159265-0.707584436725+6.283185307=8.717193520275\approx 8.72`

When
`n=2`:

`x=0.707584436725+2\pi (2)\Rightarrow 0.707584436725+12.5663706=13.273955036725\approx 13.27`

`x=\pi -0.707584436725+2\pi (2)\Rightarrow 3.14159265-0.707584436725+12.566370614=15.000378827275\approx 15.00`

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