Question

(Trigonometry) What is the y-value of the positive function when

Answer 1

In general a trigonometric function of the form:

`y=A\sin (B(x+C))+D`

has the following properties:

• A is the amplitude.

,

• C is the phase shift (positive to the left)

,

• D is the vertical shift

,

• The period is given as:

`(2\pi)/(B)`

From the information given we have that the amplitude is 3, then:

`A=3`

The period is 6pi then we have:

`\begin{gathered} 6\pi=(2\pi)/(B) \\ B=(2\pi)/(6\pi) \\ B=(1)/(3) \end{gathered}`

The horizontal shift is:

`C=-(3\pi)/(2)`

And the vertical shift is:

`D=-1`

Once we know the values we plug them in the general expression for the sine function, our function is:

`y=3\sin ((1)/(3)(x-(3\pi)/(2)))-1`

Now that we have the function we can find the its value when x=2pi, plugging this value of x in the expression we have:

`\begin{gathered} y=3\sin ((1)/(3)(2\pi-(3\pi)/(2)))-1 \\ y=3\sin ((1)/(3)((\pi)/(2)))-1 \\ y=3\sin ((\pi)/(6))=-1 \\ y=3((1)/(2))-1 \\ y=(3)/(2)-1 \\ y=(1)/(2) \end{gathered}`

Therefore, the value of y for the given x is 1/2