Question

Find the two consecutive asymptotes, the period (in radians), the phase shift (in radians) and the vertical shift of the function: y = 4csc(5θ - 3π/4)

Answer 1

Phase shift =
`(3\pi )/(5)`

Vertical shift =
`0`

Explanation:

Shift Definition

`\text{For\:}f\left(x\right)=A\cdot g\left(Bx-C\right)+D\text{,\:where\:}g\left(x\right)\text{\:is\:one\:of\:the\:basic\:trig\:functions,\:}`
`(C)/(B)\text{\:is\:phase\:shift}`,
`D\text{\:is\:vertical\:shift}`

Here we have

`\left(x\right)=4\csc \left((5\theta-3\pi )/(4)\right)`

which can be written in the form
`g\left(Bx-C\right)=\csc \left((5\theta-3\pi )/(4)\right)`

So comparing the two we get
`B=(5)/(4), C=(3\pi )/(4),\:D=0`

Phase shift =
`(C)/(B) = ((3\pi )/(4))/((5)/(4))`
`= (3\pi )/(5)`

Vertical shift
`D = 0`