Mathematics: To rewrite y(t) = 2 * sin (4 pi t) + 5 * cos (4 pi t) in the form Asin(wt+phi), you must first find the amplitude. Use c1=Asin(phi) and c2=Acos(phi) along with the Pythagorean identity to solve for A (amplitude.) previous question: By plugging in for c1 and c2, the equation should shift to be: Asin(wt+phi)=Acos(phi)sin(wt)+Asin(phi)cos(wt) Use the identity sin(x+y)=sin(x)cos(y)+cos(x)sin(y)

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To rewrite y(t) = 2 * sin (4 pi t) + 5 * cos (4 pi

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05.02.2022

The equation in standard form is $y(t) = 5.390\cdot \sin (4\pi\cdot t +0.379\pi)$ .

Step-by-step explanation:

We use the following trigonometric identity:

$\sin (\alpha + \beta) = \sin \alpha\cdot \cos \beta + \cos \alpha \cdot \sin \beta$ , $\forall\,\alpha,\beta\in \mathbb{R}$ (Eq. 1)

If we have formula $y(t) = 2\cdot \sin (4\pi\cdot t)+5\cdot \cos (4\pi\cdot t)$ , by direct comparison with (Eq. 1):

$A\cdot \cos \phi = 2$ (Eq. 2)

$A\cdot \sin \phi = 5$ (Eq. 3)

Where:

- Amplitude, dimensionless.

$\phi$ - Phase angle, measured in radians.

By dividing (Eq. 3) by (Eq. 2), we get that:

$\tan\phi = \frac{5}{2}$

$\phi = \tan^{-1}\left(\frac{5}{2} \right)$

$\phi = 0.379\pi\,rad$

From (Eq. 2) we find the amplitude:

$A = \frac{2}{\cos \phi}$

$A = \frac{2}{\cos 0.379\pi}$

A = 5.390

Therefore, we find that equation in standard form is:

$y(t) = A\cdot \sin (\omega\cdot t +\phi)$ (Eq. 4)

Where $\omega$ is the angular frequency, measured in radians per second.

( A = 5.390 , $\omega = 4\pi\,\frac{rad}{s}$ , $\phi = 0.379\pi\,rad$ )

$y(t) = 5.390\cdot \sin (4\pi\cdot t +0.379\pi)$

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