Solve the problem.
-Two sound waves, producing the pressures $P _ { 1 } = 15 \sin ( 480 \pi t )$ and $P _ { 2 } = 8 \cos ( 480 \pi t )$, combine on a person's ear drum to produce the pressure $P = P _ { 1 } + P _ { 2 }$. Graph $P$ in $[ - 0.01,0.01,0.001 ]$ by $[ - 20,20,2 ]$ and use the graph to estimate values for $a$ and $k$ so that $P = a \sin ( 480 \pi t + k )$. Round $k$ to the nearest ten-thousandth, if necessary.
A) $\mathrm { a } = 17 ; \mathrm { k } = 738.8376$
B) $\mathrm { a } = 4.1 ; \mathrm { k } = 1.0808$
C) $\mathrm { a } = 17 ; \mathrm { k } = 0.49$
D) $\mathrm { a } = 4.1 ; \mathrm { k } = 738.8376$

Question

Quizplus · Accepted Answer

The combined pressure $P = P_1 + P_2 = 15\sin(480\pi t) + 8\cos(480\pi t)$ can be rewritten using the sine addition formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$. To match this form, we recognize that $\sin(480\pi t)$ and $\cos(480\pi t)$ can be thought of as $\sin A$ and $\cos A$, respectively. The amplitude $a$ can be found using the Pythagorean theorem, $a = \sqrt{15^2 + 8^2} = 17$. To find $k$, we use the relationship $\tan(k) = \frac{8}{15}$, which gives $k \approx 0.49$ radians.

Solve the Problem P1=15sin⁡(480πt)P _ { 1 } = 15 \sin ( 480 \pi t )P1​=15sin(480πt)

Solve the Problem $P _ { 1 } = 15 \sin ( 480 \pi t )$