Verify that the equation is an identity.
-Two sound waves, producing the pressures $P _ { 1 } = 4 \sin ( 400 \pi t )$ and $P _ { 2 } = 3 \cos ( 400 \pi t )$, combine on a person's ear drum to produce the pressure $\mathrm { P } = \mathrm { P } _ { 1 } + \mathrm { P } _ { 2 }$. Graph $\mathrm { P }$ in $[ - 0.01,0.01,0.001 ]$ by $[ - 20,20,2 ]$ and use the graph to estimate values for $\mathrm { a }$ and $\mathrm { k }$ so that $\mathrm { P } = \mathrm { a } \sin ( 400 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.
A) $\mathrm { a } = 2.2 ; \mathrm { k } = 0.9273$
B) $\mathrm { a } = 5 ; \mathrm { k } = 0.6435$
C) $a = 5 ; k = 808.6467$
D) $\mathrm { a } = 2.2 ; \mathrm { k } = 808.6467$

Question

Quizplus · Accepted Answer

The combined pressure $P = P_1 + P_2 = 4\sin(400\pi t) + 3\cos(400\pi t)$ can be rewritten using the sine addition formula, $\sin(A + B) = \sin A \cos B + \cos A \sin B$, to match the form $P = a\sin(400\pi t + k)$. By comparing coefficients, we find $a = 5$ and $k = 0.6435$, which corresponds to the phase shift and amplitude of the combined wave.

Verify That the Equation Is an Identity P1=4sin⁡(400πt)P _ { 1 } = 4 \sin ( 400 \pi t )P1​=4sin(400πt)

Verify That the Equation Is an Identity $P _ { 1 } = 4 \sin ( 400 \pi t )$