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

Trigonometry What Is The Yvalue Of The Positive Function When class=

Sagot :

In general a trigonometric function of the form:

[tex]y=A\sin (B(x+C))+D[/tex]

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:

[tex]\frac{2\pi}{B}[/tex]

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

[tex]A=3[/tex]

The period is 6pi then we have:

[tex]\begin{gathered} 6\pi=\frac{2\pi}{B} \\ B=\frac{2\pi}{6\pi} \\ B=\frac{1}{3} \end{gathered}[/tex]

The horizontal shift is:

[tex]C=-\frac{3\pi}{2}[/tex]

And the vertical shift is:

[tex]D=-1[/tex]

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

[tex]y=3\sin (\frac{1}{3}(x-\frac{3\pi}{2}))-1[/tex]

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:

[tex]\begin{gathered} y=3\sin (\frac{1}{3}(2\pi-\frac{3\pi}{2}))-1 \\ y=3\sin (\frac{1}{3}(\frac{\pi}{2}))-1 \\ y=3\sin (\frac{\pi}{6})=-1 \\ y=3(\frac{1}{2})-1 \\ y=\frac{3}{2}-1 \\ y=\frac{1}{2} \end{gathered}[/tex]

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

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