Express the function as a sinusoid of the form y = a sin (bx + c).
-$$y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )$$
A) $y = \sqrt { 34 } \sin \left( 5 x - \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$
B) $y = \sqrt { 34 } \sin \left( 5 x + \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$
C) $y = \sqrt { 34 } \sin \left( 5 x + \sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right\}$
D) $y = \sqrt { 34 } \sin \left( 5 x - \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 34 } } \right) \right)$

Question

Quizplus · Accepted Answer

The given function can be expressed as a single sinusoid using the formula $a \sin(x) + b \cos(x) = \sqrt{a^2 + b^2} \sin(x + \phi)$, where $\tan(\phi) = \frac{b}{a}$. Here, $a = 5$ and $b = -3$, so $\sqrt{a^2 + b^2} = \sqrt{34}$ and $\phi = -\sin^{-1}\left(\frac{3}{\sqrt{34}}\right)$. Therefore, the correct expression is $y = \sqrt{34} \sin(5x - \sin^{-1}\left(\frac{3}{\sqrt{34}}\right))$.

Express the Function as a Sinusoid of the Form Y y=5sin⁡(5x)−3cos⁡(5x)y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )y=5sin(5x)−3cos(5x)

Express the Function as a Sinusoid of the Form Y $y = 5 \sin ( 5 x ) - 3 \cos ( 5 x )$