Question

Select one or more expressions that together represent all solutions to the equation. Your answer should be indegrees.Assume n is any integer.2 sin(4x) + 6 = 5Choose all answers that apply:-150° + n. 360°B--37.5° + n. 90°-30° + n. 180o-7.5° + n. 90

Answer 1

We have to find the expression for x.

We have the relation:

`\begin{gathered} 2\cdot\sin (4x)+6=5 \\ 2\cdot\sin (4x)=5-6 \\ 2\cdot\sin (4x)=-1 \\ \sin (4x)=-(1)/(2) \\ 4x=\arcsin (-0.5) \end{gathered}`

As the value of the sine function is negative, we can conclude that the angle 4x is placed in the 3rd or 4th quadrant:

The expression for this angle in the 4th quadrant can be written as:

`\begin{gathered} 4x=\arcsin (-0.5)=-\arcsin (0.5)=-30\degree+n\cdot360\degree \\ x=(-30+360n)/(4) \\ x=-7.5+n\cdot90 \end{gathered}`

NOTE: We add n*360 to the value of 4x as all the angles that add a full circle to -30 will have the same value of the sine function.

If we express the the angle for the 3rd quadrant, we will have:

`\begin{gathered} 4x=\arcsin (-0.5)=210+n\cdot360 \\ x=(210+360n)/(4) \\ x=52.5+n\cdot90 \end{gathered}`

Then, the solutions for x are:

x = -7.5 + n*90

x = 52.5 + n*90