Express the function as a sinusoid of the form y = a sin (bx + c).
-$$y = 7 \sin x + 5 \cos x$$
A) $y = 74 \sin \left( x + \sin - 1 \left( \frac { 5 } { \sqrt { 74 } } \right) \right)$
B) $y = \sqrt { 74 } \sin \left( x + \sin - 1 \left( \frac { 5 } { \sqrt { 74 } } \right) \right)$
C) $y = 12 \sin ( 7 x + 5 )$
D) $y = \sqrt { 74 } \sin \left( x + \cos - 1 \left( \frac { 5 } { \sqrt { 74 } } \right) \right\}$

Question

Quizplus · Accepted Answer

The given function $y = 7 \sin x + 5 \cos x$ can be expressed as a single sinusoid using the formula $y = a \sin x + b \cos x = \sqrt{a^2 + b^2} \sin(x + \phi)$, where $\phi = \sin^{-1}\left(\frac{b}{\sqrt{a^2 + b^2}}\right)$. Here, $a = 7$ and $b = 5$, so $\sqrt{a^2 + b^2} = \sqrt{49 + 25} = \sqrt{74}$. Thus, the correct form is $y = \sqrt{74} \sin\left(x + \sin^{-1}\left(\frac{5}{\sqrt{74}}\right)\right)$, which matches choice B.

Express the Function as a Sinusoid of the Form Y y=7sin⁡x+5cos⁡xy = 7 \sin x + 5 \cos xy=7sinx+5cosx

Express the Function as a Sinusoid of the Form Y $y = 7 \sin x + 5 \cos x$