Solve the problem.
-In a particular situation, the pressure, P, exerted on a person's eardrum at a distance of 8 feet from the source is given by the function: $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )$, where $\mathrm { n }$ is a positive integer. Use the difference identity for cosine to simplify $\mathrm { P }$ in this situation.
A) $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } )$
B) $\mathrm { P } ( \mathrm { t } ) = \frac { 8 } { 9 } \cos ( \mathrm { ct } + 8 )$
C) $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( \mathrm { ct } ) - \frac { \mathrm { a } } { 8 } \sin ( \mathrm { ct } )$
D) $\mathrm { P } ( \mathrm { t } ) = \cos ( \mathrm { ct } )$

Question

Quizplus · Accepted Answer

The difference identity for cosine is $\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$. However, since $2n\pi$ is a multiple of a full circle (360 degrees or $2\pi$ radians), $\cos(2n\pi) = 1$ and $\sin(2n\pi) = 0$, simplifying the expression to $\cos(ct)$.

Solve the Problem P(t)=a8cos⁡(2nπ−ct)\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )P(t)=8a​cos(2nπ−ct)

Solve the Problem $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 8 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )$