Use the fundamental identities to find the value of the trigonometric function.
-Two sound waves, producing the pressures $\mathrm { P } _ { 1 } = 8 \sin ( 460 \pi \mathrm { t } )$ and $\mathrm { P } _ { 2 } = 6 \cos ( 460 \pi \mathrm { t } )$, 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 $a$ and $k$ so that $P = a \sin ( 460 \pi t + k )$. Round $k$ to the nearest ten-thousandth, if necessary.
A) $\mathrm { a } = 3.2 ; \mathrm { k } = 929.9437$
B) $a = 10 ; k = 929.9437$
C) $a = 10 ; k = 0.6435$
D) $a = 3.2 ; \mathrm { k } = 0.9273$

Question

Quizplus · Accepted Answer

Using trigonometric identities, we can rewrite $P_1$ and $P_2$ as a single sine function. The amplitude $a$ is found by $\sqrt{8^2 + 6^2} = 10$. The phase shift $k$ is found using $\tan(k) = \frac{6}{8}$, giving $k \approx 0.6435$.

Use the Fundamental Identities to Find the Value of the Trigonometric