Question #0a796

Feb 17, 2018

$3.7 \cos \left(\pi t - 5.953\right)$ i.e. the constants are $A = 3.7$, $\omega = \pi$ and $\phi = 2 \pi - {\tan}^{- 1} \left(\frac{12}{35}\right) = 5.953$

Explanation:

To express the given function in the form $A \cos \left(\omega t + \phi\right)$, notice that according to the trigonometric sum rule

$A \cos \left(\omega t + \phi\right) = A \left[\cos \left(\omega t\right) \cos \left(\phi\right) - \sin \left(\omega t\right) \sin \left(\phi\right)\right] = - A \sin \left(\phi\right) \sin \left(\omega t\right) + A \cos \left(\phi\right) \cos \left(\omega t\right)$

Comparing with the expression given, we have $\omega = \pi$ nd

$A \sin \left(\phi\right) = - 1.2 , q \quad A \cos \left(\phi\right) = 3.5$

Thus

${A}^{2} = = {\left(A \sin \left(\phi\right)\right)}^{2} + {\left(A \cos \left(\phi\right)\right)}^{2} = {\left(- 1.2\right)}^{2} + {\left(3.5\right)}^{2} = 13.69 = {3.7}^{2}$

So, $A = 3.7$

Now,
$\tan \left(\phi\right) = \frac{A \sin \left(\phi\right)}{A \cos \left(\phi\right)} = - \frac{1.2}{3.5} = - 0.3429$

Since $\sin \left(\phi\right)$ is negative while $\cos \left(\phi\right)$ is positive, the angle $\phi$ must be in the fourth quadrant. So

$\phi = 2 \pi - {\tan}^{- 1} \left(\frac{12}{35}\right) = 5.953$